# What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? Otherwise, is there any counterexample? Moreover, what conditions should be added to homotopy to get homeomorphism?

We assume additionally both X and Y are orientable.

Let $X$ be the letter

and $Y$ be the letter

Then $X$ and $Y$ are homotopy-equivalent, but they are not homeomorphic.

Sketch proof: let $f:X\to Y$ map three of the prongs of the $\mathsf{X}$ on to the $\mathsf{Y}$ in the obvious way, and let it map the fourth prong to the point at the centre. Let $g:Y\to X$ map the $\mathsf{Y}$ into those three prongs of the $\mathsf{X}$. Then $f$ and $g$ are both continuous, and $f$ is a surjection but is not injective, while $g$ is an injection but is not surjective. Now the compositions $f\circ g$ and $g\circ f$ are both easily seen to be homotopic to the identities on $X$ and $Y$, so $X$ and $Y$ are homotopy-equivalent.

In other words, observe that $\mathsf Y$ is a deformation retract of $\mathsf X$. Alternatively, observe that $\mathsf X$ and $\mathsf Y$ both retract on to the point at the centre.

On the other hand, $X$ and $Y$ are not homeomorphic. For example, removing the point at the centre of the $\mathsf{X}$ yields a space with four connected components, while removing any point from the $\mathsf{Y}$ yields at most three connected components.