Not sure if this is what you're looking for, but here are some examples:

(1) Consider $n \in \mathbb{Z}$ with prime factorization $n=\varepsilon p_1^{v_1}\dots p_k^{v_k}$, where $\varepsilon \in \{1,-1\}$ and $v_i \in \mathbb{N}$. By the chinese remainder theorem, we have a ring isomorphism
$$
\Phi : \mathbb{Z}/n \stackrel{\sim}{\longrightarrow} \prod_{i=1}^k \mathbb{Z}/(p_i^{v_i}).
$$
We can, however, consider the quotient topology $\tau$ on $\mathbb{Z}/n$, induced by the canonical projection $\pi : \mathbb{Z} \twoheadrightarrow \mathbb{Z}/n$. We see that $(\mathbb{Z}/n,\tau)$ is a discrete topological space, because $\mathbb{Z}$ is discrete. Moreover, the space $\prod_{i=1}^k \mathbb{Z}/(p_i^{v_i})$ is discrete as a product of discrete spaces. It follows that $\Phi$ is a homeomorphism.

(2) Let $f:X\to Y$ me a map between topological spaces. Denote with
$$
\Gamma(f) := \{ (x,f(x)) \mid x \in X \} \subset X \times Y
$$
the graph of $f$, equipped with the subspace topology of the product topology on $X \times Y$. Let $p_X : X \times Y \to X$, $(x,y) \mapsto x$ denote the projection in the first component. Then, it can be shown that
$$
p_X|_{\Gamma(f)} : \Gamma(f) \to X, \qquad (x,y) \mapsto x
$$
is a homeomorphism if and only if $f$ is continuous.

I hope this was helpful!