Briefly speaking, we know that a map $f$ between $2$ topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous.
So, can anyone give me $2$ counter examples(preferably simple ones) of non-homeomorphic maps $f$ between 2 topological spaces that satisfy the properties I give? (Only one of them is satisfied and 3 examples for each property.)
- $f$ is bijective and continuous, but its inverse is not continuous.
- $f$ is bijective and the inverse is continuous, but $f$ itself is not continuous.
In addition, can we think about some examples of topologies that are path-connected?
I will understand the concept of homeomorphism much better if I know some simple counterexamples. I hope you can help me out. Thanks!
Best Answer
Consider $f:[0,2\pi)\to S^1$ given by $t\mapsto\langle\cos t,\sin t\rangle,$ where $S^1$ is the unit circle in the plane, and $[0,2\pi)$ is the real interval, both considered under their appropriate subspace topology as subsets of Euclidean spaces. Then $f$ is bijective and continuous, but its inverse is not continuous, providing an example for 1. Thus, its inverse is an example for 2.
Be careful with 3, as you need to specify what you mean by "inverse" in cases where $f$ isn't bijective. An equivalent way to define homeomorphism is as a bijective, continuous, open map (maps open sets to open sets). This avoids the need to worry about inverse functions--indeed, open maps need not be injective. For an example of a surjective, continuous, open map that is not a homeomorphism (since it is not injective), consider $p:\Bbb R^2\to\Bbb R$ given by $\langle x,y\rangle\mapsto x$, where $\Bbb R^2$ and $\Bbb R$ are in their usual topologies. For an example of an injective, continuous, open map that is not a homeomorphism (since it is not surjective), consider the inclusion $(0,1)\hookrightarrow[0,1].$