If $X$ is homeomorphic to subset of set $Y$ and $Y$ is homeomorphic to subset of $X$ then are $X$ and $Y$ homeomorphic?

Question

Let $(X, \tau_1)$ and $(Y, \tau_2)$ be topological spaces. Is it true that if $X$ is homeomoprhic to a subset of $Y$ and $Y$ is homeomorphic to a subset of $X$ then $X$ and $Y$ are homeomorphic spaces?

I'm trying to find a counter-example cause I don't believe that the statement is correct, however I'm struggling since I don't know that many homeomorphisms. An idea I had is to prove that a subset of $\mathbb{R}$ is homeomorphic to a subset of $\mathbb{R^2}$ and the other way around but $\mathbb{R}$ and $\mathbb{R^2}$ are not homeomorphic.

The plane is not homeomorphic to any subset of the line. – tomasz Aug 05 '20 at 19:41 — tomasz, Aug 05 '20 at 19:41
The simplest example is probably $[0,1]$ and $(0,1)$. – tomasz Aug 05 '20 at 19:44 — tomasz, Aug 05 '20 at 19:44

score 5 · Accepted Answer · edited Aug 05 '20 at 19:45

5

$\;[0.2,\,0.7]\subseteq (0,1)=:X\;$ is homemomorphic to $\;Y:=[0,1]\;$ , and $\;(0.2,\,0.7)\subseteq Y\;$ is homeomorphic to $\;X\;$ , but ...

edited Aug 05 '20 at 19:45

Brian M. Scott

631,399

answered Aug 05 '20 at 19:43

DonAntonio

214,715

score 2 · Answer 2 · answered Aug 05 '20 at 19:52

It is even possible to find a compact Hausdorff space $X$ with the following property, though the example is a bit complicated:

If $Y$ is the space obtained from $X$ by adding a single isolated point, then $X$ and $Y$ are not homeomorphic, $X$ is homeomorphic to a subset of $Y$, and $Y$ is homeomorphic to a subspace of $X$.

See this answer and the PDF linked from it.

If $X$ is homeomorphic to subset of set $Y$ and $Y$ is homeomorphic to subset of $X$ then are $X$ and $Y$ homeomorphic?

2 Answers2