Dimension of homotopy equivalent CW complexes

Question

Let $X$ and $Y$ be $n$ and $m$-dimensional CW complexes. If $X\simeq Y$ then $n =m$? I think it's true but I don't know the reason.

Answer 1

Take $X = \{0\}$ and $Y = [0,1]$. Then $X \simeq Y$, but $\dim X = 0 < 1 = \dim Y$.