Homotopy Group is Homeomorphism Invariant

Theorem
Let $X$ and $Y$ be two topological spaces.

Let $\phi: X \to Y$ be a homeomorphism.

Let $x_0 \in X$, $y_0 \in Y$.

Then for all $n \in \N$ the induced mapping:


 * $\phi_* : \pi_n (X,x_0) \to \pi_n(Y,y_0):$


 * $\! \left[{\!\left[{\, c\,}\right]\!}\right] \mapsto \! \left[{\!\left[{\, \phi\circ c\,}\right]\!}\right]$

is an isomorphism, where $\pi_n$ denotes the $n$th homotopy group.

Proof
Let $\phi:X\to Y$ be a homeomorphism. We must show that:


 * 1) If $c:[0,1]^n \to X$ is a continuous mapping, then $ \phi \circ c:[0,1]^n \to Y$ is also continuous
 * 2) If $c,d:[0,1]^n \to X$ are freely homotopic, then $\phi\circ c, \phi\circ d:[0,1]^n \to Y$ are also freely homotopic
 * 3) If $c,d:[0,1]^n \to X$ are not freely homotopic, there can be no free homotopy between $\phi\circ c$ and $\phi \circ d$;
 * 4) The image of the concatenation of two maps, $\phi(c * d)$, is the concatenation of the images, $\phi(c)*\phi(d)$.