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  • Functions
    ==Sets and relations== '''Example (lists).''' Sets given explicitly -- as lists -- are simplest ones:
    151 KB (25,679 words) - 17:09, 20 February 2019
  • A graph, non-tree representation of the topology of a gray scale image by Saveliev
    ...ion]], respectively. The proposed method represents the hierarchy of these sets, and the [[topology]] of the image, by means of a graph. This graph contain ...The rationale for this approach is that the connected components of these sets are arguably building blocks of real items depicted in the image.
    41 KB (6,854 words) - 15:05, 28 October 2011
  • Topological spaces
    ==Open and closed sets== and other related issues, all we need is to equip each of the sets involved with an additional structure called ''topology''.
    27 KB (4,693 words) - 02:35, 20 June 2019
  • Operations on functions
    ==Operations with sets== We can form a new set that contains all the elements of the two sets.
    142 KB (23,566 words) - 02:01, 23 February 2019
  • The topology of a gray scale image
    ...ge. The boundaries of the objects are the level curves. Since all of these sets are connected collections of pixels, they will be represented as $0$- and $ Observe that all thresholds correspond to [[upper and lower level sets]].
    15 KB (2,589 words) - 12:31, 11 September 2013
  • Neighborhoods and topologies
    ...ion into a basis, we'd have to add all ''singletons'', i.e., the one-point sets, to the collection. Since those are simply balls of diameter $0$, we end up The sets of all:
    16 KB (2,758 words) - 00:19, 25 November 2015
  • Tree representation of images
    ...]. A similar data structure is created for the holes (with the upper level sets). Combined these two graphs represent the topology of the image, the [[topo The [[connected components]] of [[upper and lower level sets]] of the [[gray scale function]] are building blocks of [[image segmentatio
    8 KB (1,263 words) - 18:45, 9 February 2011
  • Relative topology
    ...he topology<!--\index{topology}--> is given by a collection $\tau$ of open sets? Then how do we set up a topology for a subset? ...dex{open sets}--> in the $x$-axis $A$ can be seen as intersections of open sets in the $xy$-plane $X$:
    34 KB (6,089 words) - 03:50, 25 November 2015
  • Manifolds
    A basis determines what sets are open in $X$. ..., we ''separate'' the two points from each other by means of disjoint open sets:
    51 KB (8,919 words) - 01:58, 30 November 2015
  • Bases of neighborhoods
    ...ion into a basis, we'd have to add all ''singletons'', i.e., the one-point sets, to the collection. Since those are simply balls of diameter $0$, we end up The sets of
    11 KB (2,025 words) - 14:57, 2 August 2014
  • Open and closed sets
    ...ds and topologies|neighborhoods]] $\gamma$ in $X$ is given. We define open sets as ones where every point has its own neighborhood: Some sets are neither closed nor open (unlike doors). Examples in in ${\bf R}$:
    4 KB (625 words) - 01:55, 1 October 2013
  • A graph representation of the topology of color images by Saveliev
    ...vel function]] of the image. The rationale for this approach is that these sets are arguably building blocks of real items depicted in the image. Now we wo ...level sets may be objects and the connected components of the lower level sets may be holes in these objects. In addition to these, in order to capture th
    6 KB (1,011 words) - 15:33, 28 October 2011
  • Continuous functions
    ...that we have developed, the '''definition''' becomes as follows. Given two sets $X,Y$ with bases of neighborhoods $\gamma_X,\gamma_Y$, a function $f:X\to Y We start with a simple observation that if we replace neighborhoods with open sets in the definition, it would still guarantee that the function is continuous
    42 KB (7,138 words) - 19:08, 28 November 2015
  • Graph representation of gray scale images
    ...es [[adjacency]]). The [[connected components]] of [[upper and lower level sets]] of the [[gray scale function]] are building blocks of [[image segmentatio ...f a real object depicted in the image cutting through upper or lower level sets. However, one can imagine a picture with bald spot merging with the sky beh
    4 KB (653 words) - 04:45, 11 February 2011
  • Triangulations
    ...hat if we aren't interested in these “small” open sets but in “large” open sets? We choose the latter to be unions of the interiors of simplices: Let $\gamma := \{U,V,W\}$ be this open cover of $X$. These sets came from the stars of the three vertices of $K$: ${\rm St}_A, {\rm St}_A,
    30 KB (5,172 words) - 21:52, 26 November 2015
  • Quotients
    ==Quotient sets== ...re we consider the topological issues, let's take care of the underlying ''sets''<!--\index{quotient set}-->.
    26 KB (4,538 words) - 23:15, 26 November 2015
  • Web search traffic
    <td class="TableCell">intersection of closed sets is closed proof</td> <td class="TableCell">the complement of a collection of closed sets is open</td>
    24 KB (3,456 words) - 13:01, 30 September 2011
  • Products
    ...to every point on the $x$-axis. Or, we can think in terms of ''products of sets'': Generally, for any two sets $X$ and $Y$, their product set is defined as the set of ordered pairs taken
    44 KB (7,951 words) - 02:21, 30 November 2015
  • Introduction to abstract mathematics: course
    *Sets: *New sets from old:
    3 KB (373 words) - 16:06, 25 September 2013
  • Nerve of cover
    ...ntation if all you have is a topological space, i.e., a collection of open sets. ...sted in "small" open sets, i.e., ones inside simplices but in "large" open sets that are unions of the interiors of simplices.
    8 KB (1,389 words) - 13:35, 12 August 2015
  • Classes of functions
    Let's consider the two sets we considered in Chapter 2: '''Definition.''' Suppose sets $X$ and $Y$ are given. A function $f:X\to Y$ is a ''constant function'', i.
    143 KB (24,052 words) - 13:11, 23 February 2019
  • Compact spaces
    '''Definition.''' For any topological space $X$, a collection of open sets $\alpha$ is called an ''open cover''<!--\index{open cover}--> if $\cup \alp '''Exercise.''' Provide a definition of compactness in terms of closed sets.
    19 KB (3,207 words) - 13:06, 29 November 2015
  • Algebra on graphs
    ...tion.''' A ''graph''<!--\index{graph}--> $G =(N,E)$ consists of two finite sets: We have been looking at the ''subsets'' of the sets of nodes and edges to study the topology of the graph. Next, we will pursue
    36 KB (6,177 words) - 02:47, 21 February 2016
  • Separation axioms
    ...om as we separate the two points from each other by means of disjoint open sets: <center>for any $x,y \in X, x \neq y$, there are open sets $U, V$ such that $x \in U, y \in V$ and $U \cap V = \emptyset$.</center>
    3 KB (620 words) - 16:49, 27 August 2015
  • Functions of several variables
    '''Theorem.''' Level sets don't intersect. of the plane is called a ''sub-level set'' of $f$. These sets are used to convert gray-scale images to binary:
    97 KB (17,654 words) - 13:59, 24 November 2018
  • Homology groups of graphs
    We have been looking at the ''subsets'' of the sets of nodes and edges to study the topology of the graph. Next, we will pursue We have seen the importance of ''subsets'' of these sets. First, combinations of nodes form components:
    28 KB (4,685 words) - 17:25, 28 November 2015
  • Inclusion tree
    Suppose you have a collection of sets that is "[[nested collection|nested]]": if two sets intersect, one contains the other.
    1 KB (167 words) - 01:28, 30 January 2011
  • Differential equations
    ==Solution sets of ODEs== Next, ODEs produce families of curves as the sets of their solutions... and vice versa: if a family of curves is given by an
    64 KB (11,426 words) - 14:21, 24 November 2018
  • A new look at continuity
    These sets are open intervals: In dimension $2$, the relation between these sets is illustrated as follows:
    17 KB (2,946 words) - 04:51, 25 November 2015
  • Category
    ...mphasizes the point that they are not functions just as the objects aren't sets. *sets with functions,
    7 KB (1,007 words) - 22:17, 18 April 2014
  • New sets from old
    [[Products of sets]]: Given two sets X and Y.
    977 bytes (182 words) - 15:01, 25 March 2010
  • Objects in gray scale images
    ...The boundaries of the objects are the [[level curves]]. Since all of these sets are connected collections of pixels they will be represented as 0- and 1-[[ ...ding the interior) are objects. These are called the lower and upper level sets.
    10 KB (1,607 words) - 23:18, 28 January 2011
  • Test
    #[[Introduction to point-set topology|Topology with points and sets only]] #[[Open and closed sets]]
    16 KB (2,139 words) - 23:01, 9 February 2015
  • Union of a finite collection of closed sets is closed
    ...collection of open sets is open]]. Closed sets are [[complement]] of open sets after all. It's a part of the definition of [[topological space]] in terms of closed sets, by the way.
    364 bytes (60 words) - 13:55, 31 October 2010
  • Vector functions
    ==Level sets== [[Image:level sets in R2.jpg|right]]
    28 KB (4,769 words) - 19:42, 18 August 2011
  • Intersection of any collection of closed sets is closed
    ...collection of open sets is open]]. Closed sets are [[complement]]s of open sets after all. It's a part of the definition of [[topological space]] in terms of closed sets, by the way.
    359 bytes (60 words) - 13:58, 31 October 2010
  • From continuity to point-set topology
    These sets are open intervals but can be also seen as "balls": So, in dimension $2$, the relation between these sets is illustrated as follows:
    7 KB (1,207 words) - 13:01, 12 August 2015
  • Computational Homology by Kaczynski, Mischaikow, Mrozek
    **2.1 [[cubical complex|Cubical Sets]] **2.2 The Algebra of Cubical Sets
    5 KB (616 words) - 14:03, 6 October 2016
  • Applied Topology and Geometry
    #[[Introduction to point-set topology|Topology with points and sets only]] #[[Open and closed sets]]
    16 KB (2,088 words) - 16:37, 29 November 2014
  • Approaches to image analysis
    ''Level sets'' ...have a single function - the gray level. Now what we do is take its level sets (or sub-level, does not matter in this context) and analyze them as binary
    13 KB (2,018 words) - 13:55, 12 May 2011
  • Gestalt and computer vision
    ...set]]s of the [[gray scale function]], as well as [[upper and lower level sets]]. In the [[blur]]red image above, the circle is still recognizable regardl ...o far, so good. Unfortunately, next the authors concentrate on upper level sets exclusively. This is a common approach. The result is that you recognize on
    4 KB (723 words) - 15:04, 9 October 2010
  • Homotopy
    What if $Y$ is the disjoint union of $m$ convex sets in ${\bf R}^n$? Will we have: And so are all spaces homeomorphic to convex sets. There others too.
    46 KB (7,846 words) - 02:47, 30 November 2015
  • Products of complexes
    ...to every point on the $x$-axis. Or, we can think in terms of ''products of sets'': Generally, for any two sets $X$ and $Y$, their ''product set'' is defined as the set of ordered pairs t
    16 KB (2,892 words) - 22:39, 18 February 2016
  • Constructions
    ==Quotient sets== ...the topological issues, let's make clear what happens to the underlying ''sets''<!--\index{quotient}--> first.
    13 KB (2,270 words) - 22:14, 18 February 2016
  • Selection votes
    ...an to $B$, he may assign: $2/3$ to $A$, $1/3$ to $B$, and $0$ to $C$. That sets up a lottery for him. By allowing no more than two non-zero weights, we lim Now, we construct a simplicial complex from this open cover. The sets become the vertices and the intersections become the edges. We let
    24 KB (3,989 words) - 01:56, 16 May 2016
  • Data made Euclidean
    A graph<!--\index{graphs}--> is pure data. It consists of two sets: We now have a collection $K$ of three sets:
    31 KB (5,219 words) - 15:07, 2 April 2016
  • Simplicial complexes
    A graph<!--\index{graph}--> is pure data. It consists of two sets: We now have a collection $K$ of three sets:
    30 KB (5,021 words) - 13:42, 1 December 2015
  • Calculus Illustrated
    *Chapter 2. Introduction to sets and functions 1 Sets and relations
    16 KB (1,933 words) - 19:50, 28 June 2021
  • Simplicial complexes and maps
    A graph<!--\index{graph}--> is pure data. It consists of two sets: We now have a collection $K$ of three sets:
    27 KB (4,625 words) - 12:52, 30 March 2016
  • The topology of data
    Initially, we confirmed the accuracy of JPlex by analyzing known data sets to determine their persistent Betti numbers. Once we understood how JPlex w ...sets that are well understood, made them noisy, and applied JPlex to these sets. A few examples that were analyzed include:
    9 KB (1,431 words) - 16:57, 20 February 2011
  • Topology Illustrated -- Index
    *[[cubical sets|cubical sets]] *[[open sets|open sets]]
    16 KB (1,773 words) - 00:41, 17 February 2016
  • Topology of Surfaces by Kinsey
    ...ive neighborhoods]] -- [[Continuity]] -- [[Compact sets]] -- [[Connected sets]] -- Applications *Open sets and [[neighborhoods]] -- Continuity, [[connectedness]], and [[compactness]
    5 KB (725 words) - 12:30, 9 September 2016
  • Introduction to Topology: Pure and Applied by Adams and Franzosa
    ** 0.3. Sets and Operations on Them ** 1.1. Open Sets and the Definition of a Topology
    3 KB (311 words) - 13:36, 26 October 2012
  • Introductory algebraic topology: review exercises
    *[[Intersection of any collection of closed sets is closed ]] *[[Is the intersection of any collection of open sets always open?]]
    9 KB (1,553 words) - 20:10, 23 October 2012
  • Social choice
    ......,U_6$ respectively. We now assign letters to the intersections of these sets. This way we classify all individuals based on which statements they suppor ...ild the nerve of this cover. The intersections become the vertices and the sets become the edges:
    47 KB (8,030 words) - 18:48, 30 November 2015
  • Functions of several variables OLD
    ...rem (Path-connectedness in ${\bf R}$).''' In ${\bf R}$, the path-connected sets are: #These sets are connected, i.e., construct a continuous function on an interval (they a
    34 KB (5,636 words) - 23:52, 7 October 2017
  • The topology of a binary image
    ...n of non-overlapping regions, connected sets of black pixels and connected sets of white pixels, that covers the whole image. The partition is achieved by ...les (cycles) in the image and the arrows correspond to inclusions of these sets.
    16 KB (2,639 words) - 15:00, 27 October 2012
  • Graph representation of images
    .... A similar data structure is created for the holes (with the [[supralevel sets]]). Combined these two graphs represent the topology of the image, the [[to
    4 KB (617 words) - 17:23, 14 August 2009
  • Voting and social choice
    '''Theorem (Existence and Uniqueness of Tally).''' Suppose we are given two sets $V$ and $O$ and suppose also that a function $f:<V>\to O$ satisfies Fungibi Finally, a condition that sets this apart from an arbitrary group and an arbitrary homomorphism. We may re
    14 KB (2,570 words) - 17:10, 26 June 2016
  • Homeomorphism
    ...$\tau$ is the Euclidean topology and $\kappa$ any topology with fewer open sets (it's “sparser”), such as the topology of rays. $\square$ ...r}-->, interior<!--\index{ interior}-->, closure<!--\index{ closure}--> of sets, and convergent and divergent<!--\index{ convergence}--> sequences, continu
    13 KB (2,168 words) - 13:09, 7 August 2014
  • Internal structure of a vector space: part 1
    ==Sets of all linear combinations, spans==
    10 KB (1,614 words) - 17:13, 22 May 2012
  • Combinatorial cell complexes
    ...mensional simplicial complex $K$ given by a list vertices and simplices as sets of vertices: In terms of these sets, every face is a triple, such as $ABC$, and every edge is a double, such as
    34 KB (5,710 words) - 22:27, 18 February 2016
  • Linear functions in Euclidean space
    ==Solution sets of linear equations==
    23 KB (3,893 words) - 04:43, 15 February 2013
  • Complement
    In topology, complements of open sets are closed (see [[Open and closed sets]]) and vice versa. [[Category:sets]]
    485 bytes (84 words) - 05:25, 19 November 2010
  • Path-connectedness
    ...rem (Path-connectedness in ${\bf R}$).''' In ${\bf R}$, the path-connected sets are: #These sets are connected, i.e., construct a continuous function on an interval (they a
    8 KB (1,315 words) - 13:15, 12 August 2015
  • Lower and upper level sets
    Or, these two sets for each T: They are also called "excursion sets" (why?).
    654 bytes (110 words) - 20:03, 6 September 2010
  • Point-set topology: exercises
    ...cal space with the discrete topology, then the only connected sets are the sets of one element. #Is the union of a collection of closed sets always closed?
    5 KB (814 words) - 16:40, 4 October 2013
  • Manifolds model a curved universe
    '''Definition:''' If every point $a \in C$ has an [[open and closed sets|open]] (in $C$) set $U$, $a \in U$, [[homeomorphic]] to ${\bf R}^1$, then $ *open sets in ${\bf R}^N$.
    10 KB (1,588 words) - 17:11, 27 August 2015
  • Projection
    Given two sets $X$ and $Y$. Then $X×Y$ is the set of all pairs $(a,b)$ of elements in $X$ *[[New sets from old]]
    347 bytes (67 words) - 16:29, 1 June 2014
  • Maps of polyhedra
    ...the size of the cells in a uniform fashion. We have measured the sizes of sets topologically in terms of open covers, i.e., whether the set is included in Now, we compare the idea of measuring sizes of sets via open covers and via the diameters.
    51 KB (9,162 words) - 15:33, 1 December 2015
  • Quotient function
    Then the [[quotient sets|quotient set]] X/~ is the set of equivalence classes {[x]: x∈X}. *[[New sets from old]]
    333 bytes (54 words) - 12:24, 12 August 2015
  • More about manifolds
    ...3: course|calculus]] (or more precisely we should be talking about level ''sets''). Let's classify the level sets of a twice [[differentiable]] function.
    9 KB (1,542 words) - 19:58, 21 January 2014
  • Quotients of vector spaces
    ...related construction based on [[equivalence relation]] and [[quotients of sets]]. '''Theorem 2.''' The [[quotient sets|quotient set]] $L/M$ is a vector space with the operations:
    6 KB (1,115 words) - 16:03, 27 August 2015
  • Equivalence relation
    In other words this is a [[partition]]. See also [[quotient sets]]. *integers as equivalence classes of finite sets;
    2 KB (238 words) - 16:40, 21 May 2013
  • Robustness of topology of digital images and point clouds by Saveliev
    ...ce]] is a measure of robustness of the homology classes of the lower level sets of this function \ELZ, \Carlsson, \CZ09, \CZ. First the image is "[[thresholding|thresholded]]". The lower level sets of the gray scale
    27 KB (4,547 words) - 04:08, 6 November 2012
  • Find the smallest set containing 1/2 and closed under addition
    ...nd the smallest set $S\subset \mathbf{R}$'' containing $\frac{1}{2}$ and [[sets closed under algebraic operations|closed under addition]]. ...2}\in S,$ (b) $S$ is closed under addition, (c) $S$ is the smallest of all sets containing $ \frac{1}{2}$ and closed under addition.
    1 KB (211 words) - 05:16, 21 February 2011
  • Hausdorff space
    ...we can "separate" any two points from each other by means of disjoint open sets: for any $x,y∈X, x≠y$, there are open sets $U, V$ such that $x∈U, y∈V$ and $U ∩ V = ∅.$
    290 bytes (51 words) - 05:19, 18 February 2011
  • Product topology
    Following this lead, for any two sets $X$ and $Y$ their [[product set]] is defined as the set of ordered pairs ta becomes more meaningful now. It isn't just about sets anymore. Both ${\bf R}$ and ${\bf R}^2$ should have [[Euclidean topology]].
    8 KB (1,339 words) - 16:53, 27 August 2015
  • Algebraic topology: course
    *sets, functions, etc, *[[Open and closed sets]] in '''R'''<sup>n</sup>
    3 KB (448 words) - 13:32, 17 March 2014
  • Topology 2: course
    *sets, functions, etc, *[[Quotient sets]], the key construction
    1 KB (175 words) - 13:33, 17 March 2014
  • Isotropy in numerical PDEs
    *a sequence of sets of vertices $V_1,...,V_n,...,$ *a sequence of sets of edges $E_1,...,E_n,...$
    10 KB (1,593 words) - 13:20, 8 April 2013
  • Homology of cell complexes: course
    *sets, functions, etc, *[[Quotient sets]], the key construction
    2 KB (243 words) - 13:34, 17 March 2014
  • Realizations of cubical complexes
    ...s such. So, we can study their openness, closedness (see [[Open and closed sets]], interior, frontier and closure (see [[Classification of points with resp ...., an algebraic entity. But even when we can go back and forth between the sets and algebra, the difference is still dramatic:
    3 KB (561 words) - 18:07, 27 August 2015
  • Ayasdi
    ...cated methods from algebraic topology and geometry to analyze massive data sets". So, he is paid by a company created to commercialize the research he fund ..., clusters constructed from subsets of S specified as the [[preimage]]s of sets in the given covering of the reference space R). "
    4 KB (561 words) - 14:46, 16 October 2011
  • Topology I -- Fall 2013 -- final exam
    ...means of disjoint open sets: for any $x,y \in X, x \neq y$, there are open sets $U, V$ such that $x \in U, y \in V$ and $U \cap V = \emptyset$. Are the spa
    2 KB (317 words) - 16:34, 10 December 2013
  • Jordan theorem
    ...xes|realization]] of a cubical complex in the plane is a [[open and closed sets|closed set]], hence its [[complement]] is open, so it's not a cubical compl ...ensional cubical complex in the plane is the [[union]] of two [[disjoint]] sets, the [[closure]]s of which are connected cubical complexes $A$ and $B$ with
    1 KB (200 words) - 09:26, 3 September 2011
  • Image
    Given a [[function]] $f:X \rightarrow Y$ between two sets, the ''image of subset $A$ of $X$ under $f$'' is the set of all outputs of [[category:sets]]
    312 bytes (59 words) - 16:55, 29 July 2012
  • Restriction
    Suppose we have sets $X$ and $Y$, a [[function]] $f: X → Y$, and a subset $A$ of $X$. Then the [[category:sets]]
    283 bytes (55 words) - 16:42, 17 March 2013
  • Binary images
    ...n of non-overlapping regions, connected sets of black pixels and connected sets of white pixels, that covers the whole image. The partition is achieved by
    10 KB (1,705 words) - 21:26, 18 July 2011
  • Topology graph
    *The graph represents the hierarchy of the lower and upper level sets of the gray level function. **[[cycles]]: both upper and lower level sets are captured by circular sequences of edges.
    10 KB (1,727 words) - 15:03, 9 October 2010
  • Linear algebra
    Suppose, A = x + S = y + T (equal as sets), where S, T are linear subspaces. Then, S = T and y &isin; A. Affine subspaces are solution sets of systems of non-homogeneous equations.
    27 KB (4,667 words) - 01:07, 19 February 2011
  • Image of function
    Also, given sets $X$ and $Y$, subset $A$ of $X$, and a function $f:X \rightarrow Y$. Then th [[category:sets]]
    249 bytes (50 words) - 03:51, 15 February 2011
  • Sequences and their limits
    For $S=[0,1]$, any number $M\ge 1$ is its upper bound. However, these sets have no upper bounds: '''Example.''' For the following sets the least upper bound is $M=3$:
    64 KB (10,809 words) - 02:11, 23 February 2019
  • Calculus Illustrated -- Notation
    *$A \cup B= \{x:\ x\in A\ \texttt{ OR }\ x\in B\}\quad$ the union of sets $A$ and $B$; ...\cap B= \{x:\ x\in A\ \texttt{ AND }\ x\in B\}\quad$ the intersection of sets $A$ and $B$;
    2 KB (438 words) - 22:34, 22 June 2019
  • Topology of graphs
    ...tion.''' A ''graph''<!--\index{graph}--> $G =(N,E)$ consists of two finite sets: ...at are known to be path-connected! One can then study the topology of such sets by means of graphs represented as discrete structures:
    25 KB (4,214 words) - 16:08, 28 November 2015
  • Linear algebra of Euclidean space
    Suppose, $A = x + S = y + T$ (equal as sets), where $S, T$ are linear subspaces. Then, $S = T$ and $y \in A$. Affine subspaces are solution sets of systems of non-homogeneous equations.
    26 KB (3,993 words) - 19:48, 26 August 2011
  • Implicit Function Theorem
    [[Image:level sets in R2.jpg|right]] [[Image:level sets in R2 not curve.jpg]]
    2 KB (400 words) - 20:29, 28 August 2011
  • Students research projects
    ...I and Cubical Homology'' Abstract: Employing magnetic resonance (MR) data sets, I will investigate the advantages of cubical homology in the examination o ..., we can associate an abstract simplicial complex whose faces are the edge sets of the graphs in the collection. A bounded degree graph complex is a simpli
    11 KB (1,674 words) - 23:20, 25 October 2011
  • Calculus of chain maps
    ...\cdot$” stands for the multiplication of real numbers and, as a result, of sets of real numbers, not the (formal) multiples of cells. The same applies to t ...t as above, “$\cdot$” stands for the multiplication of real numbers and of sets of real numbers, not the formal multiplication of cells. The same applies t
    41 KB (7,344 words) - 12:52, 25 July 2016
  • Triangulations of surfaces
    In terms of sets, every face is a triple, such as $ABC$, and every edge is a double, such as [[Category:Topology]] [[Category:Sets]]
    3 KB (505 words) - 18:17, 27 August 2015
  • Linear operators: part 1
    ...t with the general setup: $f \colon X \rightarrow Y$ [[functions]] between sets. ...\colon X \rightarrow Y$, $g \colon Y \rightarrow Z$, two functions between sets $X,Y,Z$, then their ''[[composition]]'' is
    13 KB (2,086 words) - 19:58, 27 January 2013
  • Inclusion-exclusion formula
    For sets $A,B \subset X$, [[category:sets]]
    141 bytes (26 words) - 16:53, 6 September 2013
  • The algebra of cells
    '''Exercise.''' What are the open sets of this space? '''Exercise.''' Explain -- in terms of open, closed sets, etc. -- the topological meaning of adjacency.
    34 KB (5,644 words) - 13:35, 1 December 2015
  • Can a set to be both open and closed?
    Question: Can a set to be both [[open and closed sets|open and closed]]? Why: Sets aren't like doors...
    276 bytes (42 words) - 06:36, 3 September 2011
  • Vector and complex variables
    ...e two sets are ''almost'' equal to the two axes! If we append $0$ to these sets of eigenvectors, we have the following. For $\lambda = 2$, the set is the $
    113 KB (18,750 words) - 02:33, 10 December 2018
  • New complexes from old
    ...$ given by $q(x) = [x]$ is a [[chain maps|chain map]]. Then the [[quotient sets|quotient set]] $C(K)/ \sim$ is a chain complex with the boundary operator $ ...ding topological spaces in [[New topological spaces from old]] (also [[New sets from old]]).
    2 KB (310 words) - 13:36, 19 July 2011
  • Nerve of cover of sphere
    ...of its vertices to build an open cover on the sphere. Of course, the open sets aren't triangles here but their [[complement]]s. Alternatively, the six hemispheres can serve as the open sets. The resulting simplicial complex is an [[octahedron]].
    763 bytes (118 words) - 12:22, 12 August 2015
  • Classification of points with respect to a subset
    Then an [[Open and closed sets|open set]] $U$ coincides with the set of its interior points, called the '' ...t the closure of $A$ is the intersection of a certain collection of closed sets. =>
    4 KB (703 words) - 01:55, 1 October 2013
  • The algebra of oriented cells
    <TR> <TD>Bases of chain groups are:</TD><TD colspan="3">the sets of all the $k$-cells of $K$</TD> </TR> '''Exercise.''' Represent the sets below as realizations of cubical complexes. For each of them:
    36 KB (6,395 words) - 14:09, 1 December 2015
  • Cell complex
    ...!--\index{topological space}--> from cells. Cubical sets<!--\index{cubical sets}--> are unions of cubes<!--\index{cube}--> of various dimensions and the re
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  • Vector integrals
    <center>sets: $[ a, b ] = [ b, a ]$;</center> <center>oriented sets: $[ a, b ] = -[ b, a ]$.</center>
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  • Extension
    Suppose we have sets $X$ and $Y$, a subset $A$ of $X$, and a [[function]] $f: A → Y$ on $A$. T [[category:sets]]
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  • New vector spaces from old
    Then X×Y is the [[product of sets]], set of all pairs (a,b) of elements in X and Y respectively. Then the [[quotient sets|quotient set]] X/~ is a vector space with the operation
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  • Conferences and workshops
    #Exploratory and qualitative analysis of high-dimensional data sets ...les. Other examples include arrangements of points, lines, planes, convex sets, and their intersection patterns. There are many connections to linear alg
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  • The geometry a gray scale image
    ...along with their combinations) happen to the connected components of these sets:
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  • Why "preimage" in the definition of continuity?
    ...f continuous functions that don't preserve openness of sets, closedness of sets, take open to closed, and vice versa.
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  • Filtration
    ''Filtration'' is a sequence of "nested" [[sets]], [[topological space]]s, [[cell complex]]es, [[cubical complex]]es etc: [[category:topology]] [[category:sets]]
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  • Notation
    '''Sets:''' *$A\times B := \{(a,b):a\in A,b\in B\} \quad$ the product of sets $A,B$;
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  • Constant function
    A function $f:X\rightarrow Y$ between two sets that takes only one value is called ''constant''. [[category:sets]]
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  • The gradient
    ...st ''sets'' in the plane. For example, when $f$ is constant, all the level sets are empty but one which is the whole plane:
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  • Intro to Higher Mathematics -- Spring 2018 -- midterm
    $\bullet$ '''3.''' Restate the following in terms of inclusions of sets: <center>All sets are disjoint.</center>
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  • Fundamental group
    if $\alpha(s) \in V_i$ picking the corresponding inverse of each of these sets. Suppose, for example, that we have is an open cover of $[0,1]$. These sets may be disconnected. We need connected set, i.e., intervals, to build $\til
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  • Objects and holes
    *The connected components of upper level sets are the “holes”.
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  • Cycles in images
    ...n of non-overlapping regions, connected sets of black pixels and connected sets of white pixels, that covers the whole image. The partition is achieved by
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  • The political spectrum is a circle
    [[image:political spectrum -- sets.png| center]] ...er there is at least one person who supports both. In other words, the two sets of supporters ''intersect''. We then realize that the poll data contains mu
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  • Is the intersection of any collection of open sets always open?
    ...ircumstances ''is'' it open? [[intersection of a finite collection of open sets is open|Answer]].
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  • Grayscale images
    ...n of non-overlapping regions, connected sets of black pixels and connected sets of white pixels, that covers the whole image. The partition is achieved by
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  • Parametric complexes
    First the image is thresholded. The lower level sets of the gray scale function of the image form a filtration: a sequence of th ...tting. A ''direct system'' $\{K^{n}\}=\{K^{n}:n\in Q\}$ is a collection of sets indexed by a partially ordered set $Q$ so that if $n < m$ then there is a f
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  • Disjoint sets
    Those are sets that don't intersect, i.e. have no elements in common. [[category:sets]]
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  • Topology 1: course
    *sets, functions, etc, *[[Open and closed sets]] in '''R'''<sup>n</sup>
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  • Is the union of any collection of closed sets always closed?
    ...ances ''is'' such a union closed? [[union of a finite collection of closed sets is closed|Answer]].
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  • Quotient sets
    [[Category:Sets]]
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  • Integration in dimension n
    (see [[Products of sets]]) The relation of the sets of functions that are
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  • Group theory: test 1
    #Determine whether the following sets of invertible matrices is a subgroup of $GL(n,\mathbf{R}):$ (a) matrices wi
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  • Abstract simplicial complex
    [[Category:Topology]] [[category:sets]]
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  • Stability Analysis
    ...l LINEAR, the error terms (difference between solutions with two different sets of initial conditions) grow via the same recurrence as the original terms,
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  • Is projection a quotient function?
    [[category:sets]] [[category:exercises]]
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  • Image Simplification
    ...o count the coins and ignore the small features depicted on the coins, one sets the limit of 1000 pixels for the area. The algorithm produces the simplifie
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  • Calculus II -- Fall 2014
    **quizzes: taken from the textbook's exercise sets
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  • Open set
    #REDIRECT[[open and closed sets]]
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  • The union of a finite collection of closed sets is closed
    #REDIRECT [[Union of a finite collection of closed sets is closed]]
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  • More on vector spaces
    Issue: Subsets vs subspaces, sets vs vector spaces, i.e., they have extra structure (algebraic operations).
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  • College Algebra by Sullivan
    10.1 Sets and Counting
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  • Calculus with Analytic Geometry III -- Spring 2015
    **quizzes: taken from the textbook's exercise sets
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  • Quotients of sets
    #redirect[[Quotient sets]]
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  • Internal structure of a vector space: part 2
    So, ''very'' different sets span the same vector space.
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  • Guide for contributors
    *Sets, spaces, matrices, anything "large", are. These are much less common.
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  • Acyclic models
    [[Category:topology]] [[Category:sets]]
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  • Sublevel set
    #redirect[[Lower and upper level sets]]
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  • Is the image of a closed set under a contiuous function closed?
    Question: Is the [[image of function|image]] of a [[open and closed sets|closed set]] under a [[continuous function]] always closed?
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  • Image segmentation
    *[[Level sets method]]
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  • Calculus II -- Spring 2018
    **in-class quizzes: taken from the textbook's exercise sets;
    12 KB (1,928 words) - 19:15, 12 April 2018
  • Open sets
    #REDIRECt[[Open and closed sets]]
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  • The union of any collection of open sets is open
    #REDIRECT [[Union of any collection of open sets is open]]
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  • Eilenberg–Steenrod axioms of homology
    '''2. [[Excision]]''': Cutting out open sets doesn't change the homology. That is, if $(X,A)$ is a pair and $U$ is a sub
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  • Commutative diagram
    [[category:sets]] [[category:linear algebra]]
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  • Cell counting
    ...problem using different strategies such as [[gray scale watershed]], level-sets, and [[related approaches|other methods]].
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  • Ranking votes
    '''Corollary.''' Suppose we are given two sets $V$ and $O$. Suppose also that a function $f:<V>\to O$ satisfies Fungibilit
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  • Internal structure of a vector space: part 3
    ...oof:''' Suppose not. Suppose $a_1,\ldots,a_n$ and $b_1,\ldots,b_n$ are two sets of coordinates of $v \in V$ with respect to the same basis $B$:
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  • Higher order votes
    These sets are all groups with respect to the obvious operation. They are related to e
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  • Acyclic ranking
    #If the set of candidates is split into two sets $X,Y$ with no preferences between their members, the average ranks of these
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  • Subset
    [[category:sets]]
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  • Is the image of an open set under a continuous function open?
    Question: Is the [[image of function|image]] of a [[open and closed sets|open set]] under a [[continuous function]] always open?
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  • Inclusion
    [[Category:Sets]]
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  • Calculus I -- Fall 2016
    **in-class quizzes: taken from the textbook's exercise sets
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  • Parametrized complexes
    ...ce'' is a measure of robustness of the homology classes of the lower level sets of this function.
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  • Thresholding
    Observe that thresholds correspond to [[upper and lower level sets]].
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  • Equilibria of dynamical systems
    ...bout ''continuous dynamical systems'', i.e., systems that come as solution sets of [[ODEs]]?
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  • Compact-open topology
    ...logy coincides with the topology of [[uniform convergence]] on [[compact]] sets. If $X$ is a compact space, this is the topology of uniform convergence.
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  • Cell maps
    ...rips all structure off a given category and leaves us with the category of sets. For example, for groups we have:
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  • Real analysis: final 2
    ...nces, (b) in terms of $\varepsilon -\delta$ (c) in terms of open or closed sets. Prove that if $A\subset S_{1}$ is connected then so is $f(A).$
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  • Intersection of a finite collection of open sets is open
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  • Homology as an equivalence relation
    between such sets is explained in calculus: it is understood via its coordinate functions
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  • Advanced Calculus II -- Spring 2017 -- midterm
    $\bullet$ '''6.''' Prove that the union of two compact sets is compact.
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  • Supralevel sets
    #redirect[[Lower and upper level sets]]
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  • JPlex
    *JPlex has memory issues with even relatively small data sets when analyzing higher dimensions (four and up).
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  • Closed
    #REDIRECT[[open and closed sets]]
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  • Calculus I -- Fall 2017
    **in-class quizzes: taken from the textbook's exercise sets;
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  • Partition
    [[Category:Sets]]
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  • Topological invariants
    '''Metatheorem.''' ''Every property expressed in terms of open sets only is a topological invariant.''
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  • Euler and Lefschetz numbers
    '''Theorem (Inclusion-Exclusion Formula).''' For sets $A,B \subset X$, we have:
    41 KB (7,169 words) - 14:00, 1 December 2015
  • Compactness
    ...X$ (or a subset $X$ of some other topological space), a collection of open sets $\alpha$
    4 KB (635 words) - 12:57, 12 August 2015
  • Chain complexes
    '''Exercise.''' Represent the sets below as realizations<!--\index{realization}--> of cubical complexes. In or
    33 KB (5,293 words) - 03:06, 31 March 2016
  • Relation
    [[category:sets]]
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  • Intro to Higher Mathematics -- Spring 2016 -- midterm
    $\bullet$ '''3.''' Restate the following in terms of inclusion of sets:
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  • Homology classes
    between such sets is explained in calculus: it is understood via its coordinate functions
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  • Analysis of sample images
    ...ins and ignore the noise and the small features depicted on the coins, one sets the lower limit for the area at 1000 pixels.
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  • Surfaces and their homology
    In terms of sets, every face is a triple, such as $ABC$, and every edge is a double, such as
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  • Lefschetz number
    ...air moves within this layer. Assuming that this layer is [[open and closed sets|closed]] (does not gradually comes to nothing), what we have is a "thick" [
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  • Closed set
    #REDIRECT[[open and closed sets]]
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  • Calculus I -- Spring 2017
    **in-class quizzes: taken from the textbook's exercise sets
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  • Persistence
    ...[[gray scale image]]. Its elements (called [[frames]]) are the lower level sets of the [[gray scale function]] of the image.
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  • Topologically closed
    ...ncept equivalent to the usual definition closed set, see [[Open and closed sets]].
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  • Euler characteristic
    This well-known "[[inclusion-exclusion formula]]" for sets, $A,B \subset X$,
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  • Compliment
    [[category:sets]]
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  • Chain operators
    where $C_k(K)$ and $C_k(L)$ are the sets of all $k$-chains in $K$ and $L$ respectively. If such a function also pres
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  • Riemann integral
    The relation of the sets of functions that are
    2 KB (361 words) - 16:47, 30 September 2013
  • Introduction to Topology by Gamelin and Greene
    1 [[Open and closed sets]]
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  • Homology in Calculus
    But if we want to combine these sets into one, we realize that, every time, the constants might be different! T
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  • Applications of integral calculus
    Representations of sets of all points $1$ unit away from the origin are presented below, for the ''
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  • Systems of ODEs
    Thus systems of ODEs produce families of curves as the sets of their solutions. Conversely, if a family of curves is given by an equati
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  • Level set
    [[Image:level sets in R2.jpg|right]]
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  • Closed subset
    #REDIRECT[[open and closed sets]]
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  • Calculus Illustrated -- preface
    ...not, however, repetitive (unless include “repeat”). Additional exercise ''sets'' are at the end of each part as well as online. Keep in mind that these ex
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  • Point-set topology: course
    *[[Open and closed sets]] in '''R'''<sup>n</sup>
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  • Union of any collection of open sets is open
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  • Exponential identity of functions
    [[category:sets]]
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  • Computing persistent homology of filtrations
    *The user sets a threshold $p$ for persistence and the $p$-noise group of the filtration i
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  • Change of variables in vector spaces
    [[category:sets]] [[category:linear algebra]]
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  • Set-valued maps
    '''Exercise.''' Provide a formula for this map. Examine preimages of open sets under this map.
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  • Inverse
    [[category:sets]]
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  • Homology maps
    where $C_k(K)$ and $C_k(L)$ are the sets of all $k$-chains in $K$ and $L$ respectively. If such a function also pres
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  • Applied Calculus -- Spring 2015
    **quizzes: taken from the textbook's exercise sets
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  • The algebra of cubical cells
    '''Exercise.''' Represent the sets below as realizations of cubical complexes. For each of them:
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  • Links
    *Exercise sets
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  • Cochains on graphs
    Next, with ordered sets of nodes $N$ and edges $E$ fixed as bases of the groups of chains, both cha
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  • Calculus Two by Flanigan and Kazdan
    *0.2 [[Sets]] 1
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  • Precalculus with Scientific Applications -- Spring 2016
    **quizzes: taken from the textbook's exercise sets
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  • Upper and lower level sets
    #REDIRECT[[Lower and upper level sets]]
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  • Extrema of functions of several variables
    [[Image:open and closed sets.jpg|right]] For more see [[Open and closed sets]].
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  • Cubical complex
    '''Exercise.''' Represent the sets below as realizations<!--\index{realization}--> of cubical complexes. For e
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  • Sign
    The relation of the sets of functions that are
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  • Invertible
    [[category:sets]]
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  • Homology of maps
    ...rips all structure off a given category and leaves us with the category of sets. For example, for groups we have:
    41 KB (6,926 words) - 02:14, 21 October 2015
  • Bijection
    [[category:sets]]
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  • New topological spaces from old
    ...ace $X$ and an [[equivalence relation]] $\sim$ on $X$. Then the [[quotient sets|quotient set]] $X/ \sim$ becomes a topological space with the [[quotient sp
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  • The algebra of oriented cubes
    '''Exercise.''' Represent the sets below as realizations of cubical complexes. For each of them:
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  • Lower level set
    #REDIRECT[[Lower and upper level sets]]
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  • College Algebra -- Fall 2011
    *Quizzes: taken from the exercise sets, open book
    3 KB (355 words) - 22:00, 12 December 2011
  • Calculus exercises: advanced
    *For the following functions $F$ and sets $D$, sketch $F(D)$:
    14 KB (2,538 words) - 18:35, 14 October 2017
  • Preimage
    [[category:sets]]
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  • Vector calculus: exam 4
    1. For the following functions $F$ and sets $D$, sketch $F(D)$:
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  • Forms in Euclidean spaces
    Our main conclusion is that the “difference” between the sets of closed and exact forms reveals the topology of the domain (and vice vers
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  • Cubical complexes
    '''Exercise.''' Represent the sets below as realizations<!--\index{realization}--> of cubical complexes. In or
    20 KB (3,319 words) - 14:18, 18 February 2016
  • Simply connected spaces
    '''Exercise:''' Verify whether the following sets are simply connected:
    5 KB (785 words) - 22:07, 3 January 2014
  • Invertible function
    [[category:sets]] [[category:linear algebra]]
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  • Homology of parametric images by Saveliev
    ...n [[image processing]]. One of the reasons is that homology theory studies sets, i.e., [[binary images]]. Meanwhile, images typically seen in practical app
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  • Bijective
    [[category:sets]]
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  • One-to-one function
    [[category:sets]]
    127 bytes (21 words) - 05:29, 22 February 2011
  • The high contrast homology of a gray scale image
    ...d the lowest gray level within the object and the opposite for upper level sets. This measurement is purely topological. Also, unlike the other measurement
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  • Metric space
    The topology on $X$ generated by these sets (as a [[basis of topology]]) is called the ''[[metric topology]]''. This to
    819 bytes (155 words) - 05:11, 18 February 2011
  • College Algebra -- Fall 2013
    *Quizzes: taken from the exercise sets, open book
    6 KB (752 words) - 04:19, 13 December 2013
  • Calculus of sequences
    '''Example (spreadsheet).''' For larger sets of data, we use a spreadsheet. Recall the formulas.
    113 KB (18,425 words) - 13:42, 8 February 2019
  • Products of vector spaces
    ...[[vector space]]s L and M. Define the ''product'' L×M as the [[product of sets]], set of all pairs (a,b) of elements in X and Y respectively, as a vector
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  • Vector fields
    ...and space, numbers and vectors are supplied and placed in each of the four sets of columns of our spreadsheet one row at a time:
    91 KB (16,253 words) - 04:52, 9 January 2019
  • Geometric complexes
    ...$-additivity: for all countable collections $\{a_i\}$ of pairwise disjoint sets in $\Sigma$ we have:
    20 KB (3,354 words) - 17:37, 30 November 2015
  • De Rham cohomology
    The main idea is that the "difference" between the sets of closed and exact forms reveals the topology of the domain (and vice vers
    6 KB (938 words) - 20:55, 13 March 2013
  • Singular chain complexes
    ...jects that we call "''[[cell]]s''". The collection $K$ is partitioned into sets of cells of various "''[[dimension]]s''" $Q_k,k=0,1,2,...$. They are unders
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  • Is a set a subset of itself?
    [[category:sets]][[category:exercises]] [[category:lazy bum exercises]]
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  • Homotopy and homotopy equivalence
    What if $Y$ is the disjoint union of $m$ convex sets in ${\bf R}^n$? Will we have: And so are all spaces homeomorphic to convex sets. There others too.
    45 KB (7,738 words) - 15:18, 24 October 2015

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