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• Compact spaces
  Since there is, at most, one point of $A$ per element of $\alpha$, this new collection can't cover the whole $A$ because $A$ is infinite. A contradiction. '''Definition.''' For any topological space $X$, a collection of open sets $\alpha$ is called an ''open cover''<!--\index{open cover}-->
  19 KB (3,207 words) - 13:06, 29 November 2015
• The topology of a gray scale image
  ...rsect, one contains the other (cf. [[Nested Interval Theorem]]). Then, the collection can be represented by its ''inclusion tree'': an arrow goes from a set to a A simple example is the collection of all intervals $(-x,x)$. Its inclusion tree has a single branch.
  15 KB (2,589 words) - 12:31, 11 September 2013
• Inclusion tree
  Suppose you have a collection of sets that is "[[nested collection|nested]]": In this case, the collection can be represented by its ''inclusion [[tree]]'': an arrow goes from a set
  1 KB (167 words) - 01:28, 30 January 2011
• Limits and continuity
  ...precisely, we will construct a sequence of ''nested intervals''<!--\index{nested intervals}--> By the ''Nested Intervals Theorem'', the sequences converge to the same values:
  107 KB (18,743 words) - 17:00, 10 February 2019
• Persistent homology
  [[Filtration]] is a sequence of "nested" [[cell complex]]es, where the arrows are [[inclusion]]s: ...e methods come from ''[[homology theory]]''. One starts by considering the collection $C_{k}(K)$ of all combinations of cells of the same dimension $k,$ called '
  9 KB (1,504 words) - 04:14, 6 November 2012
• Vietoris-Rips complex
  ...string of 100 numbers or simply a vector of dimension 100. The result is a collection of disconnected 1000 points, called a [[point cloud]], in the 100-dimension Together, they form a sequence called [[filtration]] as a sequence of "nested" [[simplicial complex]]es:
  4 KB (630 words) - 20:15, 27 August 2015
• Homology maps
  ...mply means that $g$ maps $k$-chains to $k$-chains. More precisely $g$ is a collection of functions: A ''[[filtration]]'' is a sequence of "nested" complexes:
  13 KB (2,315 words) - 22:15, 15 July 2014