"closure preserves homeomorphism"

Question

Let me explain the title of the problem and the problem very clearly :

If $X$ and $Y$ are subsets of a topological spaces $A$ and $B$ respectively, which are homeomorphic in the respective subspace topology, does it imply that their closure $\bar{X}$ and $\bar{Y}$ are homeomorphic ?

Does this answer your question? Do homeomorphisms preserve closures of subspaces? — BCLC, Jun 11 '24 at 21:01

score 10 · Accepted Answer · answered Nov 09 '13 at 18:13

10

Hint. Consider $\mathbb{R}$ and $(0, 1)$.

answered Nov 09 '13 at 18:13

Karolis Juodelė

9,778

Not really a hint, per se.... – Nov 09 '13 at 20:35

score 1 · Answer 2 · answered Nov 09 '13 at 20:33

1

To the edited question, the answer is no. Consider $Y=X=A=(0,1)$, $B=[0,1]$.

answered Nov 09 '13 at 20:33

Martin Argerami

217,281

score 1 · Answer 3 · answered Nov 09 '13 at 20:36

1

For another mode of failure....

Let $A$ and $B$ the the Euclidean plane. Let $X$ be the unit interval $(0,1)$ on the $x$-axis. Let $Y$ be the unit circle minus the point $(-1,0)$.

answered Nov 09 '13 at 20:36

"closure preserves homeomorphism"

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