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• Simplicial complexes
  ...geometric simplices defined by them. We will refer by the same name to the union of these simplices. Topological spaces homeomorphic to geometric simplicial ...The ''boundary''<!--\index{boundary}--> of a geometric $n$-simplex is the union of all of its $(n-1)$-faces.
  30 KB (5,021 words) - 13:42, 1 December 2015
• The algebra of cells
  '''Exercise.''' Show that the union of the bases of all open cells in the Euclidean space ${\bf R}^N$ form its *if the cell $\sigma$, or the union of cells $\sigma := \cup _i \sigma _i$, is thought of as a ''subset'' of th
  34 KB (5,644 words) - 13:35, 1 December 2015
• Union of any collection of open sets is open
  Prove that the [[union]] of any collection of [[open set]]s is open. ...opology]] that lies inside. Take that neighborhood. It will lie inside the union set.
  445 bytes (73 words) - 13:58, 31 October 2010
• Data made Euclidean
  ...geometric simplices defined by them. We will refer by the same name to the union of these simplices. Topological spaces homeomorphic to geometric simplicial ...The ''boundary''<!--\index{boundary}--> of a geometric $n$-simplex is the union of all of its $(n-1)$-faces.
  31 KB (5,219 words) - 15:07, 2 April 2016
• Chain complexes of cell complexes
  The $0$-[[skeleton]] $K^{(0)}$ is defined as the [[disjoint union]] of $0$-cells, as points: ...)$-skeleton $K^{(m+1)}$. More precisely, it is defined as the [[disjoint]] union of the $m$-skeleton $K^{(m)}$ and all the $(m+1)$-cells, under a certain [[
  7 KB (1,225 words) - 14:05, 4 August 2013
• Is the complement of a linear subspace always a linear subspace?
  ...about the [[Is the union of two linear subspaces always a linear subspace?|union]]? the [[Is the intersection of two linear subspaces always a linear subspa
  444 bytes (65 words) - 23:30, 18 November 2010
• Notation
  *$X \sqcup Y \quad$ the disjoint union of $X,Y$; *$X \vee Y := \left(X \sqcup Y \right) /\{p\} \quad$ the one-point union of spaces $X,Y$;
  8 KB (1,519 words) - 16:30, 1 December 2015
• Is the intersection of two linear subspaces always a linear subspace?
  ...about the [[Is the union of two linear subspaces always a linear subspace?|union]]? the [[Is the complement of a linear subspace always a linear subspace?|c
  613 bytes (96 words) - 23:35, 18 November 2010
• Point-set topology: exercises
  #Is the union of a collection of closed sets always closed? ...ine a collection of subsets of $A$ as $τ_A = \{W∩A: W∈τ\}$. Prove that the union of any subcollection of $τ_{A}$ belongs to $τ_{A}$.
  5 KB (814 words) - 16:40, 4 October 2013
• Maps of polyhedra
  while the open star is the union of the insides of all these cells: *(d) The union of the equator and a meridian of the torus ${\bf T}^2$ is a deformation ret
  51 KB (9,162 words) - 15:33, 1 December 2015
• Simply connected spaces
  [[image:union of simply connected.png|center]] [[image:union of simply connected-paths.png|center]]
  5 KB (785 words) - 22:07, 3 January 2014
• Modelling motion on manifolds
  Recall that if we take all parametric curves through $a$ in $M$, then the union of all the tangent vectors at $a$ they produce is a [[vector space]], $T_aM the disjoint union of all tangent spaces.
  2 KB (377 words) - 17:13, 27 August 2015
• Surface
  '''Theorem.''' The disjoint union of two surfaces is a surface. *the union of a finite number of circles.
  5 KB (718 words) - 18:16, 27 August 2015
• Nerve of cover
  ...that this is a subcomplex of $K$. We will also use the word "star" for the union of ${A}$ and the interiors of all the simplices that contain $A$: ...plicial complex $K$ and a simplex $C$ in $K$ define the star of $C$ as the union of the interiors of all simplices in $K$ that contain $C$ (interior of a ve
  8 KB (1,389 words) - 13:35, 12 August 2015
• Surfaces and their homology
  '''Theorem.''' The disjoint union of two surfaces is a surface. *the union of a finite number of circles.
  17 KB (2,696 words) - 00:47, 12 January 2014
• Euler characteristic of graphs
  '''Exercise.''' Suppose graph $G$ is the disjoint union<!--\index{disjoint union}--> of $m$ trees, find its Euler characteristic.
  11 KB (1,876 words) - 19:23, 10 February 2015
• Topological spaces
  Of course, condition (T3') implies that the union of any ''finite'' collection of closed sets is closed. '''Proof.''' We want to show that the complement of the union of the interior and exterior consists of all points that are limit points o
  27 KB (4,693 words) - 02:35, 20 June 2019
• Applications of integral calculus
  ...t how the (unsigned) lengths of intervals behave is that the length of the union of two intervals is the sum of the two lengths minus the lengths of the int In other words, the area of the union of two regions is the sum of the two areas minus the area of the intersecti
  103 KB (18,460 words) - 01:01, 13 February 2019
• Topology Illustrated -- Index
  *[[disjoint union|disjoint union]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Calculus and algebra vs topology
  Suppose $R$ is the union of $m$ disjoint open intervals, $I_1,...,I_m$ in ${\bf R}$. '''Theorem.''' If the domain $R$ is the union of $m$ disjoint open intervals, then
  4 KB (598 words) - 21:26, 8 February 2013

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