- #1
mynameisfunk
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Hey guys, sorry for practically flooding the forum today but I have an analysis exam and nobody is more helpful than phys forum folk.
I am having trouble understanding a line in Rudin. Thm 2.36:
If {[tex]K_{\alpha}[/tex]} is a collection of compact subsets of a metric space X s.t. the intersection of every finite subcollection of {[tex]K_{\alpha}[/tex]} is nonempty, then [tex]\bigcap K_{\alpha}[/tex] is nonempty.
pf:
Fix a member [itex]K_1[/itex] of {[itex]K_{\alpha}[/itex]} and put [itex]G_{\alpha}=K^{c}_{\alpha}[/itex]. Assume that no point of [itex]K_1[/itex] belongs to every [itex]K_{\alpha}[/itex]. Then the sets [itex]G_{\alpha}[/itex] form an open cover of [itex]K_1[/itex]; and since [itex]K_1[/itex] is compact, there are finitely many indices [itex]\alpha_1,...,\alpha_n[/itex] such that [itex]K_1 \subset G_{\alpha_{1}} \cup... \cup G_{\alpha_{n}}[/itex]. But this means that [itex]K_1 \cap K_{\alpha_{2}} \cap... \cap K_{\alpha_{n}}[/itex] is empty, in contradiction to our hypothesis.
I do not see how this proves anything. We are assuming the opposite of the theorem is true and we arrive at the opposite of our result, but that shouldn't really constitute a proof should it?
I am having trouble understanding a line in Rudin. Thm 2.36:
If {[tex]K_{\alpha}[/tex]} is a collection of compact subsets of a metric space X s.t. the intersection of every finite subcollection of {[tex]K_{\alpha}[/tex]} is nonempty, then [tex]\bigcap K_{\alpha}[/tex] is nonempty.
pf:
Fix a member [itex]K_1[/itex] of {[itex]K_{\alpha}[/itex]} and put [itex]G_{\alpha}=K^{c}_{\alpha}[/itex]. Assume that no point of [itex]K_1[/itex] belongs to every [itex]K_{\alpha}[/itex]. Then the sets [itex]G_{\alpha}[/itex] form an open cover of [itex]K_1[/itex]; and since [itex]K_1[/itex] is compact, there are finitely many indices [itex]\alpha_1,...,\alpha_n[/itex] such that [itex]K_1 \subset G_{\alpha_{1}} \cup... \cup G_{\alpha_{n}}[/itex]. But this means that [itex]K_1 \cap K_{\alpha_{2}} \cap... \cap K_{\alpha_{n}}[/itex] is empty, in contradiction to our hypothesis.
I do not see how this proves anything. We are assuming the opposite of the theorem is true and we arrive at the opposite of our result, but that shouldn't really constitute a proof should it?