Real Analysis – Open Interval vs Half-Open Interval Homeomorphism

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As the topic, how can I prove it?

Prove that (a,b) and [a,b) in R are not homeomorphic metric space.

I'm pretty weak on this section (homeomorphism) and I don't have any clue to figure it out.

Please help me, thank you.

Best Answer

Without going into formalisms, if two spaces are homeomorphic then all of their topological properties are the same. Now the space $(a,b)$ has the (topological) property that if you remove any of its points, the resulting space is disconnected. But the space $[a,b)$ does not have this property -- why?

Best Answer

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