Consider [0,1], [0,1), and (0,1) as subspaces of R with the standard topology. Prove that none of these spaces are homeomorphic.

Hint: try removing points and see what happens with cutsets.

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- Mar 18th 2009, 07:52 PMAndreametHoeomorphic topological spaces
Consider [0,1], [0,1), and (0,1) as subspaces of R with the standard topology. Prove that none of these spaces are homeomorphic.

Hint: try removing points and see what happens with cutsets. - Mar 19th 2009, 04:55 PMaliceinwonderland
Based on your hint, suppose we have a homeomorphism between [0,1] and [0,1) such that

.

If h is homeomorphism, then the restriction of h removing 1 from the domain of h and h(1) from the codomain should be homeomorphism such that

.

We see that is not a homeomorphism because the domain of is connected but the codomain of is not connected. Contradiction.

Thus, h is not a homeomorphism either.

The remaining cases are similar to the above one.