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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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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

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