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Homework Statement
This one has been bothering me for a while.
One needs to show that [0, 1]ω is not compact in the uniform topology.
The Attempt at a Solution
As a reminder, the uniform topology on Rω is induced by the uniform metric, which is defined with d(x, y) = sup{min{|xi - yi|, 1} : i = 1, 2, ...}.
For any ε > 1, and for any x in Rω, the open ball B(x, ε) equals Rω.
Now, clearly, for any x, y in [0, 1]ω, d(x, y) = sup{|xi - yi|}.
The main problem seems to be that I can't figure out what kind of sets (except finite sets) are compact in Rω (if at all)?