Difference between revisions of "Compact times paracompact implies paracompact"

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==Facts used==
 
==Facts used==
  
# [[uses::Tube lemma]]: If <math>X</math> is a compact space and <math>Y</math> is a topological space. Then, given any open subset <math>U</math> of <math>X \times Y</math> containing <math>X \times \{ y \}</math> for some <math>y \in Y</math>, there exists an open subset <math>V</math> of <math>Y</math> such that <math>y \in V</math> and <math>X \times V \subseteq U</math>.
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# [[uses::Tube lemma]]: Suppose <math>X</math> is a compact space and <math>Y</math> is a topological space. Then, given any open subset <math>U</math> of <math>X \times Y</math> containing <math>X \times \{ y \}</math> for some <math>y \in Y</math>, there exists an open subset <math>V</math> of <math>Y</math> such that <math>y \in V</math> and <math>X \times V \subseteq U</math>.
  
 
==Proof==
 
==Proof==
  
'''Given''': A compact space <math>X</math>, a paracompact space <math>Y</math>.
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'''Given''': A compact space <math>X</math>, a paracompact space <math>Y</math>. <math>\{ U_i \}_{i \in I}</math> form an open cover of <math>X \times Y</math>.
  
'''To prove''' If <math>U_i</math> form an open cover of <math>X \times Y</math>, there exists a locally finite open refinement of the <math>U_i</math>.
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'''To prove''': There exists a locally finite open refinement of the <math>U_i</math>s, i.e., an open cover <math>\{ Q_k \}_{k \in K}</math> of <math>X \times Y</math> such that:
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* It is locally finite: or any point <math>(x,y) \in X \times Y</math>, there exists an open set <math>R</math> containing <math>(x,y)</math> that intersects only finitely many of the <math>Q_k</math>s.
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* It refines <math>\{ U_i \}_{i \in I}</math>: Every <math>Q_k</math> is contained in one of the <math>U_i</math>s.
  
 
'''Proof''':
 
'''Proof''':
  
# 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>: 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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{| class="sortable" border="1"
# Let <math>W_y</math> be the union of this finite collection of open subsets <math>U_i</math>. By fact (1), there exists an open subset <math>V_y</math> of <math>Y</math> such that <math>X \times V_y \subseteq W_y</math>.
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! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
# The <math>V_y</math> form an open cover of <math>Y</math>.
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# There exists a locally finite open refinement, say <math>\mathcal{P}</math> of the <math>V_y</math> in <math>Y</math>: This follows from the fact that <math>Y</math> is paracompact.
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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.
# We can construct a locally finite open refinement of <math>U_i</math> from these:
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|-
## For each member <math>P \in \mathcal{P}</math>, there exists <math>V_y</math> such that <math>P \subseteq V_y</math>. Thus, <math>X \times P \subseteq X \times V_y \subseteq W_y</math>. <math>W_y</math>, in turn, is a union of a finite collection of <math>U_i</math>s. Thus, <math>X \times P</math> is the union of the intersections <math>(X \times P) \cap U_i</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>.
## Since the <math>X \times P</math> together cover <math>X \times Y</math>, the <math>(X \times P) \cap U_i</math> are an open cover of <math>X \times Y</math> that refines the <math>U_i</math>s.
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## Finally, we argue that <math>(X \times P) \cap U_i</math> is a ''locally finite'' open cover: Suppose <math>(x,y) \in X \times Y</math>. Since <math>\mathcal{P}</math> is a locally finite open cover of <math>Y</math>, there exists an open subset <math>Q</math> of <math>Y</math> containing <math>y</math> such that <math>Q</math> intersects only finitely many members of <math>\mathcal{P}</math>. Thus, the neighborhood <math>X \times Q</math> intersects only finitely many <math>X \times P</math>s, which in turn give rise to finitely many <math>(X \times P) \cap U_i</math>s each. Thus, <math>X \times Q</math> intersects only finitely many members of the open cover.
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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>.
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|-
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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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|-
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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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|-
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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 P_j</math>, and for any <math>j<math>, there are finitely many <math>k \in K</math> with <math>k = (i,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 the <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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|-
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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 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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|}

Revision as of 16:25, 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

Verbal statement

The product of a compact space with a paracompact space (given the product topology), is paracompact

Statement with symbols

Let X be a compact space and Y a paracompact space. Then X \times Y 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: or 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) Since P_js refine V_ys (Step (4)), there exists y \in Y such that P_j \subseteq V_y. In turn, by the definition of V_y (Step (2)), we have X \times V_y \subseteq W_y, which in turn is a union of finitely many U_is (Step (1)). Thus, X \times P_j is contained in a union of finitely many U_is, and hence, is the union of its intersection with those U_is. Since U_i are all open, the intersections (X \times P_j) \cap U_i are all open.
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 P_j, and for any j<math>, there are finitely many <math>k \in K with k = (i,j). Steps (4), (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 the (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), (6) Since \{ P_j \}_{j \in J} form a locally finite open cover of Y (Step (4)), there exists an open subset S of Y such that S contains y and S intersects only finitely many of the P_js. Set R  = X \times S, so R is open in X \times Y. R therefore intersects only finitely many of the X \times P_js. For any Q_k, with k = (i,j), we have Q_k \subseteq P_j by construction (Step (6)), so if Q_k intersects R so does X \times P_j. Thus, the Q_ks that intersect R must correspond to the finitely many js for which R intersects X \times P_j. Since there are finitely many ks for each j, this gives that there are finitely many Q_ks intersecting R.