Rasalhague
Dec3-10, 09:29 PM
I'm trying to understand the proof given in the last 10 minutes or so of this (http://www.youtube.com/watch?v=AQHVdiXRXQA&feature=channel) video lecture, but after some struggle, it occurs to me that I may be misinterpreting what the theorem says. According to this, Cantor's finite intersection principle states the following.
Given a metric space (X,d) and a collection of compact subsets
\left \{ K_\alpha \subseteq X \; \bigg| \; \alpha \in A \right \}
(where A is an index set), if the elements of any finite subcollection of \left \{ K_\alpha \right \}_{\alpha \in A} have a nonempty intersection, then the intersection of all the K_\alpha is nonempty.
But should "any" be read in the specific sense, "there exists", here:
\left [ \exists \left \{ K_{\alpha_i} \right \}_{i=1}^n \subseteq \left \{ K_\alpha \right \}_{\alpha \in A} \left \left ( \bigcap_{i=1}^{n} K_{\alpha_i} \neq \varnothing \right ) \right ] \Rightarrow \left [ \bigcap_{\alpha \in A} K_\alpha \neq \varnothing \right ] \enspace ?
Or should "any" be read in the nonspecific sense, "for all":
\left [ \forall \left \{ K_{\alpha_i} \right \}_{i=1}^n \subseteq \left \{ K_\alpha \right \}_{\alpha \in A} \left \left ( \bigcap_{i=1}^{n} K_{\alpha_i} \neq \varnothing \right ) \right ] \Rightarrow \left [ \bigcap_{\alpha \in A} K_\alpha \neq \varnothing \right ] \enspace ?
And what sense or senses could collection (http://en.wikipedia.org/wiki/Collection_%28mathematics%29) have here: set, class, family, multiset?
Given a metric space (X,d) and a collection of compact subsets
\left \{ K_\alpha \subseteq X \; \bigg| \; \alpha \in A \right \}
(where A is an index set), if the elements of any finite subcollection of \left \{ K_\alpha \right \}_{\alpha \in A} have a nonempty intersection, then the intersection of all the K_\alpha is nonempty.
But should "any" be read in the specific sense, "there exists", here:
\left [ \exists \left \{ K_{\alpha_i} \right \}_{i=1}^n \subseteq \left \{ K_\alpha \right \}_{\alpha \in A} \left \left ( \bigcap_{i=1}^{n} K_{\alpha_i} \neq \varnothing \right ) \right ] \Rightarrow \left [ \bigcap_{\alpha \in A} K_\alpha \neq \varnothing \right ] \enspace ?
Or should "any" be read in the nonspecific sense, "for all":
\left [ \forall \left \{ K_{\alpha_i} \right \}_{i=1}^n \subseteq \left \{ K_\alpha \right \}_{\alpha \in A} \left \left ( \bigcap_{i=1}^{n} K_{\alpha_i} \neq \varnothing \right ) \right ] \Rightarrow \left [ \bigcap_{\alpha \in A} K_\alpha \neq \varnothing \right ] \enspace ?
And what sense or senses could collection (http://en.wikipedia.org/wiki/Collection_%28mathematics%29) have here: set, class, family, multiset?