# Homeomorphic Spaces in Topology

Let $$X$$ and $$Y$$ be two topological spaces. Let $$f : X \to Y$$ and $$g: Y \to X$$, such that

1. $$f$$ and $$g$$ are surjective;
2. $$f$$ and $$g$$ are continous.

Does this imply that $$X$$ and $$Y$$ are homeomorphic?

It seems similar to Bernstein's theorem in set theory, and many of the topological properties like compactness, connectedness, etc are getting preserved.

Any help would be appreciated.

• You should ask a different question. Editing an already answered question changing the question altoghether disrupts the question system and disregards the work which was already put in the answers below. – Jackozee Hakkiuz May 13 at 20:54
• @JackozeeHakkiuz Apologies. I'll do that. – Kr Dpk May 13 at 20:55
• @JackozeeHakkiuz math.stackexchange.com/q/4138074/499096 – Kr Dpk May 13 at 20:59

No. There exists a continuous surjection $$f$$ between $$\mathbb{R}$$ and $$\mathbb{R}^2$$ (see here), and a continuous surjection $$g$$ between $$\mathbb{R}^2$$ and $$\mathbb{R}$$ (obvious), but they are not homeomorphic.

• Thanks :). A follow up question, what would happen if f and g were taken to be bijective ? – Kr Dpk May 13 at 20:43
• For a continuous surjection $$S^1 \mapsto [-1,+1]$$ take the first coordinate projection map $$(x,y) \mapsto x$$.
• For a continuous surjection $$[-1,+1] \mapsto S^1$$ take the map $$x \mapsto (\cos(2 \pi x),\sin(2\pi x))$$.

No it does not.

Consider $$[0,1]$$ and $$[0,1]^2$$. The projection on any component certainly constitutes a continuous surjection $$[0,1]^2 \rightarrow [0,1]$$.

Conversely, there are space filling curves, giving continuous surjections $$[0,1]\rightarrow [0,1]^2$$.

But $$[0,1]$$ and $$[0,1]^2$$ are not homeomorphic. Deleting a point in the interior of $$[0,1]$$ results in a disconnected space, which is not true, when deleting a point in the interior of $$[0,1]^2$$.

• Thanks for replying. A follow up question, what would happen if f and g were taken to be bijective ? – Kr Dpk May 13 at 20:45
• That’s a good one. – PrudiiArca May 13 at 20:48