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• Compact spaces
  ==Open covers and accumulation points== Then, for any point $x \in X$, there is an open neighborhood $U_x$ of $x$ such that
  19 KB (3,207 words) - 13:06, 29 November 2015
• Triangulations
  ...x{cubical complex}--> may be thought of as ''open'', i.e., homeomorphic to open balls, while the cells in cell (and simplicial) complexes are ''closed'', i ...ell complex but only a topological space, i.e., a set with a collection of open subsets? What topological spaces are polyhedra?
  30 KB (5,172 words) - 21:52, 26 November 2015
• Nerve of cover
  ...presentation if all you have is a topological space, i.e., a collection of open sets. ...nterested 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
• Manifolds
  ...--\index{locally Euclidean space}--> $n$ if for every $x\in X$ there is an open set $U$ such that $x\in U$ and there is a homeomorphism $h:{\bf R}^n \to U$ ...se, “homeomorphic to ${\bf R}^n$” can be replaced with “homeomorphic to an open $n$-ball”, or “box”, etc.:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Maps of polyhedra
  while the open star is the union of the insides of all these cells: ...en stars of all vertices of complex $K$ forms an open cover<!--\index{open cover}--> of its realization $|K|$.
  51 KB (9,162 words) - 15:33, 1 December 2015
• Compactness
  ...ace $X$ (or a subset $X$ of some other topological space), a collection of open sets $\alpha$ is called an ''open cover'' if $\cup \alpha = X$ (or $X \subset \cup \alpha$).
  4 KB (635 words) - 12:57, 12 August 2015
• Products
  $$\{U \times V : U \text{ open in } X,\ V \text{open in } Y \}.$$ ...to work, we need to show that compactness holds even if we only deal with open covers of a particular kind.
  44 KB (7,951 words) - 02:21, 30 November 2015
• Topology Illustrated -- Index
  *[[open balls|open balls]] *[[open cell|open cell]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Forms in Euclidean spaces
  ...ddition, calculus would be incomplete unless we are able to limit it to an open subset $U$ of ${\bf R}^n$. Now, what if we are to represent an open subset $U$ of ${\bf R}^n$ as a realization of a cell complex? Since such a
  44 KB (7,778 words) - 23:32, 24 April 2015
• Nerve of cover of sphere
  ...em.''' Find an open cover of the sphere '''S'''² the [[nerve of cover|nerve]] of which is homeomorphic to '''S'''². ...ars]] of its vertices to build an open cover on the sphere. Of course, the open sets aren't triangles here but their [[complement]]s.
  763 bytes (118 words) - 12:22, 12 August 2015
• Fundamental group
  So, we start with an [[open cover]] of the circle of, say, two elements: [[image:cover of circle.png|center]]
  10 KB (1,673 words) - 18:23, 2 December 2012
• Separation axioms
  ..." axiom 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 $U \cap V = \emptyset$.</cente
  3 KB (620 words) - 16:49, 27 August 2015
• A new look at continuity
  ...to find the area $A$ of each of them in order to know how many we need to cover the whole floor. These sets are open intervals:
  17 KB (2,946 words) - 04:51, 25 November 2015
• Systems of ODEs
  ...t)$ and $y=y(t)$ (a parametric curve) with either one differentiable on an open interval $I$ such that for every $t$ in $I$ we have: for every $t$ in some open interval that contains $t_0$.
  63 KB (10,958 words) - 14:27, 24 November 2018
• Selection votes
  ...olves it, however, for a few very special cases only. Thus the problem was open and well-known at the time. Eckmann decided to look at it from the viewpoin Now, we construct a simplicial complex from this open cover. The sets become the vertices and the intersections become the edges. We le
  24 KB (3,989 words) - 01:56, 16 May 2016
• Introductory algebraic topology: review exercises
  *[[Can a set to be both open and closed? ]] *[[Is the intersection of any collection of open sets always open?]]
  9 KB (1,553 words) - 20:10, 23 October 2012
• Homotopy
  ...$f^{-1}(D)\subset (0,1)$ is open, and, therefore, is the disjoint union of open intervals. Pick one of them, $(a,b)$. Then we have: This construction only works when there are finitely many such open intervals.
  46 KB (7,846 words) - 02:47, 30 November 2015
• Vector fields
  It is as if we cover the whole stream with those little balls and study their rotation. ...rm the following function of two variables to study this further (as if we cover the whole stream with those little balls).
  91 KB (16,253 words) - 04:52, 9 January 2019
• Homotopy and homotopy equivalence
  ...$f^{-1}(D)\subset (0,1)$ is open, and, therefore, is the disjoint union of open intervals. Pick one of them, $(a,b)$. Then we have: This construction only works when there are finitely many such open intervals.
  45 KB (7,738 words) - 15:18, 24 October 2015
• Approximating paths
  '''Theorem.''' In an [[open]] region $R \subset {\bf R}^n$, if two points are connected by a [[path]], We cover the path between $A$ and $B$ with disks $D_1,...,D_n$, within $R$, and then
  1 KB (248 words) - 20:58, 7 February 2013

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