Example:

The torus defined above is homeomorphic to $S^1\times S^1$. To see this, let $T$ denote the torus and $\eta:I^2\longrightarrow T$ be the quotient map. Define the map $f:I^2\longrightarrow S^1\times S^1$ as

$$f(x, y) = (\exp(2\pi ix), \exp(2\pi iy)).$$

There is a unique bijection $\overline{f}:T\longrightarrow S^1\times S^1$ such that $\overline{f}\circ \eta = f.$ The universal property shows that $\overline{f}$ is continuous and since $T$ is compact and $S^1\times S^1$ is Hausdorff, the map ${\overline f}$ is a closed map and so a homeomorphism.