Difference between revisions of "Compact times paracompact implies paracompact"

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(Proof)
 
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! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
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! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation !! Commentary
 
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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. || Begin focusing on a slice.
 
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| 2 || For any point <math>y \in Y</math>, let <math>W_y</math> be the union of the 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 the 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>. || Use compactness to get a tube around the slice.
 
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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>. 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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| 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>. 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>. || Project down to the paracompact space.
 
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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. || Use paracompactness to get the locally finite open refinement, in the projected-down setting.
 
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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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| 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> || Go back to the big space, reversing the projection step.
 
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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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| 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>. || Wrap things up.
 
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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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| 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> || Wrap things up.
 
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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.
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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. || --
 
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{{tabular proof format}}
 
{{tabular proof format}}

Latest revision as of 16:59, 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: Note that in the proof below, Step (4) has three aspects (locally finite, cover, and refinement) and different later steps use different aspects of Step (4), with the specific aspect used indicated parenthetically in the Previous steps used column.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation Commentary
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. Begin focusing on a slice.
2 For any point y \in Y, let W_y be the union of the 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. Use compactness to get a tube around the slice.
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. Since \bigcup_{y \in Y} \{ y \} = Y, and y \in V_y \subseteq Y, we get \bigcup_{y \in Y} V_y = Y. Project down to the paracompact space.
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. Use paracompactness to get the locally finite open refinement, in the projected-down setting.
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] Go back to the big space, reversing the projection step.
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. Wrap things up.
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] Wrap things up.
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. --
This proof uses a tabular format for presentation. Learn more about tabular proof formats|View all pages on facts with proofs in tabular format