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• 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

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