Given $(a,b)$, and $(c,d) \cup (h,k)$ subspaces of $\mathbb{R}$ with the absolute value metric induced topology where $c
We have just started working with homeomorphisms and I am not sure of cut and dry ways to show that these two spaces are not homeomorphic. We haven't discussed connectedness or covering properties as of when the question was stated.
I am guessing that the spaces are not homeomorphic and I assume it has something to do with $(a,b)$ cannot be constructed by the union of disjoint open intervals but clearly $(c,d) \cup (h,k)$ is such a construction. I am not sure what kind of topological differences this would imply as this is fairly new material and as far as I know the ways that we have to disprove two spaces are homeomorphic are kind of limited with what we have been shown so far.
I do see that a bijection can definitely be constructed, I would imagine, between the spaces but I am thinking though that the spaces are not homeomorphic.
Any advice? =/