Prove that an open interval (0.1) and a closed interval [0,1] are not homeomorphic.
I am trying to prove this statement but only vague ideas on how to start.
Not using connectedness properties.
Please help
[Math] Prove that an open interval (0,1) and a closed interval [0,1] are not homeomorphic.
general-topologygeometric-topology
Best Answer
Consider the sequence $\left(\frac1n\right)_{n\in\mathbb N}$ in $(0,1)$. It has no subsequence which converges to an element of $(0,1)$. However, every sequence of elements of $[0,1]$ has a subsequence that converges to an element of $[0,1]$, by the Bolzano-Weierstrass theorem and because $[0,1]$ is closed. Therefore, $(0,1)$ and $[0,1]$ are not homeomorphic.