If there are continuous surjections $f: X \to Y$ and $g: Y \to X$, are $X, Y$ homeomorphic?

Question

If there are continuous surjections $f: X \to Y$ and $g: Y \to X$ are $X, Y$ homeomorphic? This is not homework. Just to trying to review some topology.

Ah, it looks like this is a duplicate: http://math.stackexchange.com/q/169468/264 — Zev Chonoles, Feb 25 '13 at 18:41

score 5 · Answer 1 · answered Feb 25 '13 at 18:40

5

Hint: Let $X=\mathbb{S}^1$ and $Y=[0,1]$.

answered Feb 25 '13 at 18:40

Zev Chonoles

129,973

score 1 · Answer 2 · edited Apr 13 '17 at 12:58

1

See for even stronger examples the answers to https://mathoverflow.net/questions/30661/non-homeomorphic-spaces-that-have-continuous-bijections-between-them

edited Apr 13 '17 at 12:58

Community

1

answered Feb 25 '13 at 19:04

Henno Brandsma

242,131

If there are continuous surjections $f: X \to Y$ and $g: Y \to X$, are $X, Y$ homeomorphic?

2 Answers2