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

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

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

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

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