Difference between revisions of "Compact times paracompact implies paracompact"

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(Proof)
(Proof)
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! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
 
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
 
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| 1 || For any point <math>y \in Y</math>, there is a finite collection of <math>U_i</math> that cover <math>X \times \{ y \}</math> || ||  <math>X</math> is compact || || Since <math>X</math> is compact, the subspace <math>X \times \{ y \}</math> of <math>X \times Y</math> is also compact, so the cover by the open subsets <math>U_i</math> has a finite subcover.
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| 1 || For any point <math>y \in Y</math>, there is a finite collection of <math>U_i</math> that cover <math>X \times \{ y \}</math>. || ||  <math>X</math> is compact || || Since <math>X</math> is compact, the subspace <math>X \times \{ y \}</math> of <math>X \times Y</math> is also compact, so the cover by the open subsets <math>U_i</math> has a finite subcover.
 
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| 2 || For any point <math>y \in Y</math>, let <math>W_y</math> be the union of this finite collection of open subsets <math>U_i</math> as obtained in Step (1). There exists an open subset <math>V_y</math> of <math>Y</math> such that <math>y \in V_y</math> and <math>X \times V_y \subseteq W_y</math> || Fact (1) || <math>X</math> is compact || Step (1) || Follows from Fact (1), setting the <math>U</math> of Fact (1) to be <math>W_y</math>.
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| 2 || For any point <math>y \in Y</math>, let <math>W_y</math> be the union of this finite collection of open subsets <math>U_i</math> as obtained in Step (1). There exists an open subset <math>V_y</math> of <math>Y</math> such that <math>y \in V_y</math> and <math>X \times V_y \subseteq W_y</math>. || Fact (1) || <math>X</math> is compact || Step (1) || Follows from Fact (1), setting the <math>U</math> of Fact (1) to be <math>W_y</math>.
 
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| 3 || The open subsets <math>V_y, y \in Y</math> obtained in Step (2) form an open cover of <math>Y</math>. || || || Step (2) || By Step (2), <math>y \in V_y</math>, hence <math>X \times \{ y \} \subseteq X \times V_y</math>. Since <math>\bigcup_{y \in Y} \{ y \} = Y</math>, and <math>y \in V_y \subseteq Y</math>, we get <math>\bigcup_{y \in Y} V_y = Y</math>.
 
| 3 || The open subsets <math>V_y, y \in Y</math> obtained in Step (2) form an open cover of <math>Y</math>. || || || Step (2) || By Step (2), <math>y \in V_y</math>, hence <math>X \times \{ y \} \subseteq X \times V_y</math>. Since <math>\bigcup_{y \in Y} \{ y \} = Y</math>, and <math>y \in V_y \subseteq Y</math>, we get <math>\bigcup_{y \in Y} V_y = Y</math>.
 
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| 4 || There exists a locally finite open refinement <math>\{ P_j \}_{j \in J}</math> of the <math>V_y</math> in <math>Y</math> || || <math>Y</math> is paracompact || Step (3) || Step-given combination direct.
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| 4 || There exists a locally finite open refinement <math>\{ P_j \}_{j \in J}</math> of the <math>V_y</math> in <math>Y</math>. || || <math>Y</math> is paracompact || Step (3) || Step-given combination direct.
 
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| 5 || For each <math>P_j</math>, <math>X \times P_j</math> is a union of finitely many intersections <math>(X \times P_j) \cap U_i</math>, all of which are open subsets || || || Steps (1) (2), (4) || Since <math>P_j</math>s refine <math>V_y</math>s (Step (4)), there exists <math>y \in Y</math> such that <math>P_j \subseteq V_y</math>. In turn, by the definition of <math>V_y</math> (Step (2)), we have <math>X \times V_y \subseteq W_y</math>, which in turn is a union of finitely many <math>U_i</math>s (Step (1)). Thus, <math>X \times P_j</math> is contained in a union of finitely many <math>U_i</math>s, and hence, is the union of its intersection with those <math>U_i</math>s. Since <math>U_i</math> are all open, the intersections  <math>(X \times P_j) \cap U_i</math> are all open.
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| 5 || For each <math>P_j</math>, <math>X \times P_j</math> is a union of finitely many intersections <math>(X \times P_j) \cap U_i</math>, all of which are open subsets. || || || Steps (1), (2), (4) (refinement aspect) || <toggledisplay>Since <math>P_j</math>s refine <math>V_y</math>s (Step (4)), there exists <math>y \in Y</math> such that <math>P_j \subseteq V_y</math>. In turn, by the definition of <math>V_y</math> (Step (2)), we have <math>X \times V_y \subseteq W_y</math>, which in turn is a union of finitely many <math>U_i</math>s (Step (1)). Thus, <math>X \times P_j</math> is contained in a union of finitely many <math>U_i</math>s, and hence, is the union of its intersection with those <math>U_i</math>s. Since <math>U_i</math> are all open, the intersections  <math>(X \times P_j) \cap U_i</math> are all open.</toggledisplay>
 
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| 6 || The open subsets of the form <math>(X \times P_j) \cap U_i</math> of Step (5) form an open cover of <math>X \times Y</math> that refines the <math>U_i</math>s (note that not every combination of <math>P_j</math> and <math>U_i</math> is included -- only the finitely many <math>U_i</math>s needed as in Step (5)). We will index this open cover by indexing set <math>K \subseteq I \times J</math>, and call it <math>\{ Q_k \}_{k \in K}</math>, where <math>Q_k = (X \times P_j) \cap U_i</math>. In particular, if <math>k = (i,j)</math>, then <math>Q_k \subseteq X \times P_j</math> and <math>Q_k \subseteq U_i</math>, and for any <math>j</math>, there are finitely many <math>k \in K</math> with the second coordinate of <math>k</math> equal to <math>j</math>. || || || Steps (4), (5) || <math>\{ P_j \}_{j \in J}</math> cover <math>Y</math>, so <math>\{ X \times P_j \}_{j \in J}</math> cover <math>X \times Y</math>. By Step (5), <math>X \times P_j</math> is the union of finitely many <math>(X \times P_j) \cap U_i</math>, so the latter also form an open cover of <math>X \times Y</math>.
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| 6 || The open subsets of the form <math>(X \times P_j) \cap U_i</math> of Step (5) form an open cover of <math>X \times Y</math> that refines the <math>U_i</math>s (note that not every combination of <math>P_j</math> and <math>U_i</math> is included -- only the finitely many <math>U_i</math>s needed as in Step (5)). We will index this open cover by indexing set <math>K \subseteq I \times J</math>, and call it <math>\{ Q_k \}_{k \in K}</math>, where <math>Q_k = (X \times P_j) \cap U_i</math>. In particular, if <math>k = (i,j)</math>, then <math>Q_k \subseteq X \times P_j</math> and <math>Q_k \subseteq U_i</math>, and for any <math>j</math>, there are finitely many <math>k \in K</math> with the second coordinate of <math>k</math> equal to <math>j</math>. || || || Steps (4) (cover aspect), (5) || <math>\{ P_j \}_{j \in J}</math> cover <math>Y</math>, so <math>\{ X \times P_j \}_{j \in J}</math> cover <math>X \times Y</math>. By Step (5), <math>X \times P_j</math> is the union of finitely many <math>(X \times P_j) \cap U_i</math>, so the latter also form an open cover of <math>X \times Y</math>.
 
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| 7 || The open cover <math>\{ Q_k \}_{k \in K}</math> of Step (6) is a locally finite open cover. In other words, for any <math>(x, y) \in X \times Y</math>, there is an open subset <math>R \ni (x,y)</math> such that <math>R</math> intersects only finite many <math>Q_k</math>s. || || || Steps (4), (6) || Since <math>\{ P_j \}_{j \in J}</math> form a locally finite open cover of <math>Y</math> (Step (4)), there exists an open subset <math>S</math> of <math>Y</math> such that <math>S</math> contains <math>y</math> and <math>S</math> intersects only finitely many of the <math>P_j</math>s. Set <math>R  = X \times S</math>, so <math>R</math> is open in <math>X \times Y</math>. <math>R</math> therefore intersects only finitely many of the <math>X \times P_j</math>s. For any <math>Q_k</math>, with <math>k = (i,j)</math>, we have <math>Q_k \subseteq X \times P_j</math> by construction (Step (6)), so if <math>Q_k</math> intersects <math>R</math> so does <math>X \times P_j</math>. Thus, the <math>Q_k</math>s that intersect <math>R</math> must correspond to the finitely many <math>j</math>s for which <math>R</math> intersects <math>X \times P_j</math>. Since there are finitely many <math>k</math>s for each <math>j</math>, this gives that there are finitely many <math>Q_k</math>s intersecting <math>R</math>.
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| 7 || The open cover <math>\{ Q_k \}_{k \in K}</math> of Step (6) is a locally finite open cover. In other words, for any <math>(x, y) \in X \times Y</math>, there is an open subset <math>R \ni (x,y)</math> such that <math>R</math> intersects only finite many <math>Q_k</math>s. || || || Steps (4) (locally finite aspect), (6) || <toggledisplay>Since <math>\{ P_j \}_{j \in J}</math> form a locally finite open cover of <math>Y</math> (Step (4)), there exists an open subset <math>S</math> of <math>Y</math> such that <math>S</math> contains <math>y</math> and <math>S</math> intersects only finitely many of the <math>P_j</math>s. Set <math>R  = X \times S</math>, so <math>R</math> is open in <math>X \times Y</math>. <math>R</math> therefore intersects only finitely many of the <math>X \times P_j</math>s. For any <math>Q_k</math>, with <math>k = (i,j)</math>, we have <math>Q_k \subseteq X \times P_j</math> by construction (Step (6)), so if <math>Q_k</math> intersects <math>R</math> so does <math>X \times P_j</math>. Thus, the <math>Q_k</math>s that intersect <math>R</math> must correspond to the finitely many <math>j</math>s for which <math>R</math> intersects <math>X \times P_j</math>. Since there are finitely many <math>k</math>s for each <math>j</math>, this gives that there are finitely many <math>Q_k</math>s intersecting <math>R</math>.</toggledisplay>
 
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| 8 || The open cover <math>\{ Q_k \}_{k \in K}</math> is as desired || || || Steps (6), (7) || Combine the two steps to get what we wanted to prove.
 
| 8 || The open cover <math>\{ Q_k \}_{k \in K}</math> is as desired || || || Steps (6), (7) || Combine the two steps to get what we wanted to prove.
 
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Revision as of 16:36, 3 June 2017

This article states and proves a result of the following form: the product of two topological spaces, the first satisfying the property Compact space (?) and the second satisfying the property Paracompact space (?), is a topological space satisfying the property Paracompact space (?).
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Statement

Let X be a compact space and Y a paracompact space. Then X \times Y, the Cartesian product endowed with the product topology, is paracompact.

Related facts

Other results using the same proof technique:

Facts used

  1. Tube lemma: Suppose X is a compact space and Y is a topological space. Then, given any open subset U of X \times Y containing X \times \{ y \} for some y \in Y, there exists an open subset V of Y such that y \in V and X \times V \subseteq U.

Proof

Given: A compact space X, a paracompact space Y. \{ U_i \}_{i \in I} form an open cover of X \times Y.

To prove: There exists a locally finite open refinement of the U_is, i.e., an open cover \{ Q_k \}_{k \in K} of X \times Y such that:

  • It is locally finite: For any point (x,y) \in X \times Y, there exists an open set R containing (x,y) that intersects only finitely many of the Q_ks.
  • It refines \{ U_i \}_{i \in I}: Every Q_k is contained in one of the U_is.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 For any point y \in Y, there is a finite collection of U_i that cover X \times \{ y \}. X is compact Since X is compact, the subspace X \times \{ y \} of X \times Y is also compact, so the cover by the open subsets U_i has a finite subcover.
2 For any point y \in Y, let W_y be the union of this finite collection of open subsets U_i as obtained in Step (1). There exists an open subset V_y of Y such that y \in V_y and X \times V_y \subseteq W_y. Fact (1) X is compact Step (1) Follows from Fact (1), setting the U of Fact (1) to be W_y.
3 The open subsets V_y, y \in Y obtained in Step (2) form an open cover of Y. Step (2) By Step (2), y \in V_y, hence X \times \{ y \} \subseteq X \times V_y. Since \bigcup_{y \in Y} \{ y \} = Y, and y \in V_y \subseteq Y, we get \bigcup_{y \in Y} V_y = Y.
4 There exists a locally finite open refinement \{ P_j \}_{j \in J} of the V_y in Y. Y is paracompact Step (3) Step-given combination direct.
5 For each P_j, X \times P_j is a union of finitely many intersections (X \times P_j) \cap U_i, all of which are open subsets. Steps (1), (2), (4) (refinement aspect) [SHOW MORE]
6 The open subsets of the form (X \times P_j) \cap U_i of Step (5) form an open cover of X \times Y that refines the U_is (note that not every combination of P_j and U_i is included -- only the finitely many U_is needed as in Step (5)). We will index this open cover by indexing set K \subseteq I \times J, and call it \{ Q_k \}_{k \in K}, where Q_k = (X \times P_j) \cap U_i. In particular, if k = (i,j), then Q_k \subseteq X \times P_j and Q_k \subseteq U_i, and for any j, there are finitely many k \in K with the second coordinate of k equal to j. Steps (4) (cover aspect), (5) \{ P_j \}_{j \in J} cover Y, so \{ X \times P_j \}_{j \in J} cover X \times Y. By Step (5), X \times P_j is the union of finitely many (X \times P_j) \cap U_i, so the latter also form an open cover of X \times Y.
7 The open cover \{ Q_k \}_{k \in K} of Step (6) is a locally finite open cover. In other words, for any (x, y) \in X \times Y, there is an open subset R \ni (x,y) such that R intersects only finite many Q_ks. Steps (4) (locally finite aspect), (6) [SHOW MORE]
8 The open cover \{ Q_k \}_{k \in K} is as desired Steps (6), (7) Combine the two steps to get what we wanted to prove.