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# Help... Im not sure how to do this!

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Let X be a topological space and F a collection of closed subsets. Define G as the complements of sets in F. Then a union of sets in G is the complement of an intersection of the corresponding sets in F. So 2 F has FIP if and only if G has no finite subcover of X. Also F has the total intersection property.

Thank you!!

Apr 17, 2022

#1
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Do you want the proof of this statement?

Or do you think this statement is wrong and you want to disprove it?

Apr 17, 2022
#2
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I didn't expect a topology question on this site haha

Anyways, this statement looks correct to me, and I will attempt to prove it.

Let $$\mathcal F=\{F_1, F_2, \cdots\}$$ and $$\mathcal G = \{F_1^c, F_2^c, \cdots\}$$.

(=>) Suppose that F has F.I.P. That means for any finite subset $$\{F_1', F_2', \cdots, F_N'\} \subseteq \mathcal F$$$$\displaystyle \bigcap_{i = 1}^N F_i'\neq \varnothing$$.

Further suppose on the contrary that G has a finite subcover of X, i.e., there exists a finite subset $$\{\left(F_1'\right)^c,\left(F_2'\right)^c,\cdots,\left(F_N'\right)^c\} \subseteq \mathcal G$$ where $$\displaystyle\bigcup_{i = 1}^N \left(F_i'\right)^c=X$$. Taking complement on both sides, that means there exists a finite subset $$\{F_1', F_2', \cdots, F_N'\} \subseteq \mathcal F$$ such that $$\displaystyle \bigcap_{i = 1}^N F_i' = \varnothing$$. That contradicts the F.I.P. of F. Therefore, by contradiciton, G has no finite subcover of X.

(<=) Suppose that G has no finite subcover of X. That means for any finite subset $$\{\left(F_1'\right)^c,\left(F_2'\right)^c,\cdots,\left(F_N'\right)^c\} \subseteq \mathcal G$$, we have $$\displaystyle\bigcup_{i = 1}^N \left(F_i'\right)^c\neq X$$. Again, taking complement on both sides yields for any for any finite subset $$\{F_1', F_2', \cdots, F_N'\} \subseteq \mathcal F$$, we have $$\displaystyle \bigcap_{i = 1}^N F_i' \neq \varnothing$$. Then F has F.I.P.

That finishes the proof that F has F.I.P. if and only if G has no finite subcover of X.

MaxWong  Apr 17, 2022
edited by MaxWong  Apr 17, 2022
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Would you mind explaining the definition of total intersection property in your textbook? I couldn't find a definition anywhere on the web.

Apr 17, 2022