- #1
Rasalhague
- 1,387
- 2
I'm trying to understand the proof given in the last 10 minutes or so of this 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 [itex](X,d)[/itex] and a collection of compact subsets
[tex]\left \{ K_\alpha \subseteq X \; \bigg| \; \alpha \in A \right \}[/tex]
(where [itex]A[/itex] is an index set), if the elements of any finite subcollection of [itex]\left \{ K_\alpha \right \}_{\alpha \in A}[/itex] have a nonempty intersection, then the intersection of all the [itex]K_\alpha[/itex] is nonempty.
But should "any" be read in the specific sense, "there exists", here:
[tex]\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 ?[/tex]
Or should "any" be read in the nonspecific sense, "for all":
[tex]\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 ?[/tex]
And what sense or senses could collection have here: set, class, family, multiset?
Given a metric space [itex](X,d)[/itex] and a collection of compact subsets
[tex]\left \{ K_\alpha \subseteq X \; \bigg| \; \alpha \in A \right \}[/tex]
(where [itex]A[/itex] is an index set), if the elements of any finite subcollection of [itex]\left \{ K_\alpha \right \}_{\alpha \in A}[/itex] have a nonempty intersection, then the intersection of all the [itex]K_\alpha[/itex] is nonempty.
But should "any" be read in the specific sense, "there exists", here:
[tex]\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 ?[/tex]
Or should "any" be read in the nonspecific sense, "for all":
[tex]\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 ?[/tex]
And what sense or senses could collection have here: set, class, family, multiset?