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• The union of a finite collection of closed sets is closed
  #REDIRECT [[Union of a finite collection of closed sets is closed]]
  67 bytes (11 words) - 13:55, 31 October 2010
• The union of any collection of open sets is open
  #REDIRECT [[Union of any collection of open sets is open]]
  58 bytes (10 words) - 13:58, 31 October 2010
• Union of a finite collection of closed sets is closed
  Prove that the [[union]] of a finite collection of [[closed set]]s is closed.
  364 bytes (60 words) - 13:55, 31 October 2010
• 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
• Is the union of any collection of closed sets always closed?
  Q: Is the [[union]] of any collection of [[closed set]]s always closed? *Can such a union be ''[[open set|open]]''?
  362 bytes (57 words) - 09:25, 3 September 2011
• Is the union of two linear subspaces always a linear subspace?
  Question: Is the [[union]] of two [[linear subspace]]s always a linear subspace?
  432 bytes (64 words) - 23:29, 18 November 2010

Page text matches

• Cell complex
  ...--\index{topological space}--> -- as the disjoint union<!--\index{disjoint union}-->. That's the $0$-skeleton $K^{(0)}$ of $K$. Next, we take this space $K^{(0)}$ and combine it, again as the disjoint union, with all $1$-cells in $K$. To put them together, we introduce an equivalen
  40 KB (6,459 words) - 23:27, 29 November 2015
• Combinatorial cell complexes
  ...--\index{topological space}--> -- as the disjoint union<!--\index{disjoint union}-->. That's the $0$-skeleton $K^{(0)}$ of $K$. Next, we take this space $K^{(0)}$ and combine it, again as the disjoint union, with all $1$-cells in $K$. To put them together, we introduce an equivalen
  34 KB (5,710 words) - 22:27, 18 February 2016
• Relative topology
  ...from old. The second simplest is the ''disjoint union''<!--\index{disjoint union}-->. ...e topologies of $X$ and $Y$ to remain "intact" in $Z$. But just taking the union of $\tau _X \cup \tau _X$ would not produce a topology as (T1) fails!
  34 KB (6,089 words) - 03:50, 25 November 2015
• Introductory algebraic topology: review exercises
  *[[Is the union of any collection of closed sets always closed? ]] 7. Is the union of a collection of closed sets always closed?
  9 KB (1,553 words) - 20:10, 23 October 2012
• Triangulations
  ...\index{balls}-->. The reason is that a cubical complex may be built as the union of a collection of subsets of a Euclidean space, while a cell complex is bu The ''open star''<!--\index{open star}--> is the union of the insides of all these cells:
  30 KB (5,172 words) - 21:52, 26 November 2015
• Bases of neighborhoods
  What about the union? Even though it's about the union, let's try to recycle the proof for intersection. After all, we will have t '''Theorem.''' The union of two open sets is open.
  11 KB (2,025 words) - 14:57, 2 August 2014
• Is the union of any collection of closed sets always closed?
  Q: Is the [[union]] of any collection of [[closed set]]s always closed? *Can such a union be ''[[open set|open]]''?
  362 bytes (57 words) - 09:25, 3 September 2011
• Manifolds
  The third idea is to take the intersection for $U$ and the union for $V$. This is something that might work. '''Theorem.''' The disjoint union of two $n$-manifolds is an $n$ manifold.
  51 KB (8,919 words) - 01:58, 30 November 2015
• Neighborhoods and topologies
  What about the union? Let's try to recycle the proof for intersection. After all, we will have t '''Theorem.''' The union of two open sets is open.
  16 KB (2,758 words) - 00:19, 25 November 2015
• Homotopy
  What if $Y$ is the disjoint union of $m$ convex sets in ${\bf R}^n$? Will we have: ...Q$. Then $f^{-1}(D)\subset (0,1)$ is open, and, therefore, is the disjoint union of open intervals. Pick one of them, $(a,b)$. Then we have:
  46 KB (7,846 words) - 02:47, 30 November 2015
• Homotopy and homotopy equivalence
  What if $Y$ is the disjoint union of $m$ convex sets in ${\bf R}^n$? Will we have: ...Q$. Then $f^{-1}(D)\subset (0,1)$ is open, and, therefore, is the disjoint union of open intervals. Pick one of them, $(a,b)$. Then we have:
  45 KB (7,738 words) - 15:18, 24 October 2015
• Cubical complex
  '''Definition.''' The union of the cells of a given cubical complex $K$ is called its ''realization''<! ...ls. What about infinite? Hint: unlike the union of $[-1/n,1/n],\ n>0$, the union of cells doesn't produce ''new'' limit points. This kind of collection is c
  29 KB (4,800 words) - 13:41, 1 December 2015
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  The union of any collection of pixels is a subset of the [[Euclidean space|Euclidean ...position is a [[partition]] of the union of black (closed) pixels into the union of a collection of disjointed (open) cells.
  41 KB (6,854 words) - 15:05, 28 October 2011
• Chain complexes
  Next, we take this space $K^{(0)}$ and combine it, again as the disjoint union, with all $1$-cells in $K$. To put them together, we introduce an equivalen Next, we take this space $K^{(1)}$ and combine it, again as the disjoint union, with all $2$-cells in $K$. To put them together, we introduce an equivalen
  33 KB (5,293 words) - 03:06, 31 March 2016
• Operations on functions
  [[image:boys and balls -- union.png| center]] '''Definition.''' The ''union'' of any two sets $X$ and $Y$ is the set that consists of the elements that
  142 KB (23,566 words) - 02:01, 23 February 2019
• Quotients
  ...the circle above, the preimage of an arc is either an open interval or the union of two half-open intervals at the end-points. Let's consider the second exa ...of an open disk under the identification map is either an open disk or the union of two half-disks at the edge.
  26 KB (4,538 words) - 23:15, 26 November 2015
• Topology of graphs
  ...tion}--> $|G|$ of graph $G$ is a subset of the Euclidean space that is the union of the following two subsets of the space: ...irst in order to turn nodes and edges into algebraic entities, such as the union. Unfortunately, the algebra of unions is inadequate as there is no appropri
  25 KB (4,214 words) - 16:08, 28 November 2015
• Short overview
  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 (1,179 words) - 15:27, 7 January 2014
• 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
• Open and closed sets
  '''Theorem.''' ''The union of any collection of open sets is open.'' '''Theorem.''' ''The union of two closed sets is open.''
  4 KB (625 words) - 01:55, 1 October 2013
• Properties of homology groups
  [[Image:disjoint union of cell complexes.jpg|center]] ...ealizations_of_cubical_complexes|realizations]], the homology group of the union is the [[product of vector spaces|product]] of their homology groups:
  4 KB (739 words) - 12:59, 28 August 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
• Vector fields
  where $D$ is the union of the two rectangles and $\partial D$ is its boundary. We have constructed where $D$ is the union of the two rectangles and $\partial D$ is its boundary. We continue on addi
  91 KB (16,253 words) - 04:52, 9 January 2019
• Integration in dimension n
  *$Q$ is a union of several rectangles, or ...[Additivity of integral]]).''' Integration is additive with respect to the union of domains of integration.
  33 KB (5,415 words) - 05:58, 20 August 2011
• Integration of differential forms of degree 0 and 1
  '''Definition:''' Suppose the curve is the union of finitely many smooth curves, if $C = $ union of edges $e_1, \ldots, e_s$ with $\varphi(e_i)=m_i$.
  12 KB (1,906 words) - 17:44, 31 December 2012
• 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
• The algebra of cubical cells
  '''Example.''' Compute the boundary of the union of two adjacent squares:
  46 KB (7,844 words) - 12:50, 30 March 2016
• Geometry
  or their disjoint union.
  41 KB (6,928 words) - 17:31, 26 October 2015
• Introduction to discrete calculus
  The boundary of an edge is the union of its endpoints:
  40 KB (6,983 words) - 19:24, 23 July 2016
• Constructions
  ...of an open disk under the identification map is either an open disk or the union of two half-disks at the edge.
  13 KB (2,270 words) - 22:14, 18 February 2016
• Eilenberg–Steenrod axioms of homology
  '''4. Additivity''': Homology is additive. That is, if space is the disjoint union of a family of topological spaces $\{X_{\alpha}\}$:
  3 KB (476 words) - 14:08, 4 August 2013
• Advanced Topology: exercises
  ...$Q_d$ be a cube and let $X$ be its one-dimensional skeleton, that is, the union of all edges of $Q_d$. For $d = 2, 3, 4, 5, 6$, determine the number of ver
  9 KB (1,487 words) - 18:18, 9 May 2013
• The integral
  ...f the Riemann sums but to ''adding the domains of integration'', i.e., the union of the two intervals. The idea becomes especially vivid when the formula i
  66 KB (11,473 words) - 21:36, 19 January 2019

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