Vipul: Created page with '{{product computation| left = compact space| right = metacompact space| final = metacompact space}} ==Statement== ===Verbal statement=== The product of a compact space wit…'

2009-07-17T05:16:09Z

Created page with '{{product computation| left = compact space| right = metacompact space| final = metacompact space}} ==Statement== ===Verbal statement=== The product of a compact space wit…'

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{{product computation|
left = compact space|
right = metacompact space|
final = metacompact space}}

==Statement==

===Verbal statement===

The product of a [[compact space]] with a [[metacompact space]] (given the [[product topology]]), is metacompact.

===Statement with symbols===

Let <math>X</math> be a [[compact space]] and <math>Y</math> a [[metacompact space]]. Then <math>X \times Y</math> is metacompact.

==Related facts==

Other results using the same proof technique:

* [[Compact times paracompact implies paracompact]]
* [[Compact times orthocompact implies orthocompact]]
* [[Compact times Lindelof implies Lindelof]]

==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>X \times V \subseteq U</math>.

==Proof==

'''Given''': A compact space <math>X</math>, a metaacompact space <math>Y</math>.

'''To prove''' If <math>U_i</math> form an open cover of <math>X \times Y</math>, there exists a point-finite open refinement of the <math>U_i</math>.

'''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.
# 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>.
# The <math>V_y</math> form an open cover of <math>Y</math>.
# There exists a point-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.
# We can construct a point-finite open refinement of <math>U_i</math> from these:
## 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>.
## 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.
## Finally, we argue that <math>(X \times P) \cap U_i</math> is a ''point-finite'' open cover: Suppose <math>(x,y) \in X \times Y</math>. Since <math>\mathcal{P}</math> is a point-finite open cover of <math>Y</math>. Then, there exist only finitely many <math>P \in \mathcal{P}</math> such that <math>y \in P</math>. For each of these, <math>X \times P</math> corresponds to finitely many intersections <math>(X \times P) \cap U_i</math>, so the total number of open subsets containing <math>(x,y)</math> is finite.

Compact times metacompact implies metacompact - Revision history

Vipul: Created page with '{{product computation| left = compact space| right = metacompact space| final = metacompact space}} ==Statement== ===Verbal statement=== The product of a compact space wit…'