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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
• 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
• 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
• 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
• Orientation of manifolds
  Patches are open so if they intersect, they have to overlap! Indeed, if $U \cap V \neq \emptyset$ then $U \cap V$ is open. Hence on $U \cap V$ there are two coordinate systems, from $U$ and from $V
  6 KB (983 words) - 16:30, 28 January 2013
• Vector and complex variables
  '''Example.''' There are even more outcomes that the theorem doesn't cover. Recall the characteristic polynomial of the matrix $A$ of the $90$ degree for all $z$ in an open disk $D$ around $a$ and such that the series $\sum_{n=0}^{\infty}M_n$ (of r
  113 KB (18,750 words) - 02:33, 10 December 2018
• Geometry in calculus
  ...o a}f(x)=L$ if for any open [[neighborhood]] $\epsilon$ of $L$ there is an open neighborhood $\delta$ of $a$ such that $x\in\delta$ implies $f(x)\in\epsilo <center>for any $\epsilon >0$ there is open [[cover]] $\delta$ of $[a,b]$ such that $|I-R(P)|<\epsilon$ whenever all elements o
  9 KB (1,604 words) - 18:08, 27 August 2015
• Differential calculus of parametric curves
  for all $t$ within some open interval from $t=s$; second: '''Definition.''' Suppose a parametric curve $X=F(t)$ is defined on an open interval $I$ that contains $t=s$. Then the ''derivative'' of $F$ at $t=s$ i
  130 KB (22,842 words) - 13:52, 24 November 2018
• Calculus as a part of topology
  ...o a}f(x)=L$ if for any open [[neighborhood]] $\epsilon$ of $L$ there is an open neighborhood $\delta$ of $a$ such that $x\in\delta$ implies $f(x)\in\epsilo *for any $\epsilon >0$ there is open cover $\delta$ of $[a,b]$ such that $|I-R(P)|<\epsilon$ whenever all elements of
  13 KB (2,233 words) - 14:41, 20 February 2015
• Social choice
  ...ical creeds).''' The six statements below are meant to represent an open ''cover'' of the political spectrum (in the US). In other words, we assume that eve We now build 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
• Elections
  Therefore, the nerve of this cover will contain an edge between these two vertices. Then the space of choices The open cover $\alpha$ is still the collection of $U_{ij}$ but this time there are no emp
  33 KB (5,872 words) - 13:13, 17 August 2015
• The derivative
  ...ee the tangent line as an edge of a piece of paper; then this piece has to cover none of the (relevant) part of the graph: (B) If a function is constant on an open interval $I$, then its derivative is zero for all $x$ in $I$; i.e.,
  75 KB (13,000 words) - 15:12, 7 December 2018
• Singular chain complexes
  '''Theorem (Refinement).''' If $(X,A)$ is a pair and $\gamma$ is an open cover of $X$, then the chain map generated by the [[inclusion]] of the sum of sub
  8 KB (1,367 words) - 13:49, 4 August 2013
• Test
  #[[Open and closed sets]] #Approximate topological spaces: [[Nerve of cover]]
  16 KB (2,139 words) - 23:01, 9 February 2015
• From continuity to point-set topology
  These sets are open intervals but can be also seen as "balls": *$\gamma _X$ is the set of all open balls in $X={\bf R}^n$, and
  7 KB (1,207 words) - 13:01, 12 August 2015
• Conferences and workshops
  ...s are perfect for enriching courses in these areas. Some of topics we will cover include the geometry and combinatorics of polytopes, triangulations, combin ...s a tremendous amount of work going on currently, and one can easily state open problems. It also has important applications to chemistry, biochemistry and
  8 KB (1,122 words) - 02:52, 24 October 2011
• Applied Topology and Geometry
  #[[Open and closed sets]] #Approximate topological spaces: [[Nerve of cover]]
  16 KB (2,088 words) - 16:37, 29 November 2014
• Axioms of chain complexes
  ...e the chain complex. That is, if $(X,A)$ is a pair and $\gamma$ is an open cover of $X$, then the chain map generated by the [[inclusion]] of the sum of sub
  4 KB (592 words) - 14:13, 4 August 2013