I have just learned how to prove that two subspaces of a metric space are topologically equivalent. Now I am trying to learn how to prove that two subspaces are not topologically equivalent.
I have no idea how to go about doing this. Can somebody please show me how to do that for the following simple problem, so that I may apply the same method to more complicated ones?
Prove that on the real line, the intervals $[0,1]$ and $[0,1]\cup[2,3]$ are not topologically equivalent.
I would like to prove this using my book's definition of topological equivalence:
Two metric spaces $(A,d_A)$ and $(B, d_B)$ are said to be topologically equivalent if there are inverse functions $f:A\to B$ and $g: B\to A$ such that $f$ and $g$ are continuous.
So far I have learned to prove continuity using the $\delta$, $\epsilon$ definition of continuity, and I have learned about neighborhoods, limits, and open and closed sets.