(a,b) not homeomorphic to [a,b]

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SUMMARY

The discussion centers on proving that the open interval (0,1) is not homeomorphic to the closed interval [0,1]. The key approach involves removing a point from each space and analyzing their connectedness. Participants clarify that the standard topology should be assumed unless specified otherwise. Additionally, the distinction between compactness is highlighted, with one space being compact and the other not, further solidifying the conclusion.

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Homework Statement


Show that (0,1) is no homeomorphic to [0,1]


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The Attempt at a Solution



I understand the usual procedure for proving this - remove a point from each of the spaces and check for connectedness. My only question is, the topology on each space isn't specified. Do I assume it's the subspace topology or what?

In general, when given questions like this, with no topology given, do I just infer from the context of the question which topology I should use?
 
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Nevermind guys. Looks like I need to pay more attention. My book says to assume the standard topology unless otherwise noted.
 
I understand the usual procedure for proving this - remove a point from each of the spaces and check for connectedness.

This is a pretty awesome way of proving they're not homeomorphic to be honest. I think most people would just show that one is compact and the other is not
 

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