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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
• 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
• 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
• 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

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