Unformatted text preview: f n : [0 , 1] → [0 , 1] , f n ( x ) = the n th digit in the binary expansion of x. ) 6. Prove that [0 , 1] ω ⊂ R ω is not compact in the uniform topology. Prove also that it is not compact in the box topology. (It is compact in the product topology by Tychonoﬀ’s Theorem, but you do not have to show that.) Challenge (extra credit): Let X be a locally compact topological space. Does it follow that every x ∈ X has a neighborhood whose closure is compact?...
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 Spring '05
 SMITH
 Math, Topology, Compact space, real line, product topology, compact topological space, Tychonoff

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