I was reading one of the answer on MSE: $\mathbb{R}^2/(\mathbb{Z}\times\mathbb{Z})$ homeomorphic to $S^1\times S^1$?
One corollary used is Assume that $(Y,\pi)$ is a quotient of the topological space $X$ modulo $R$. Then, for any topological space $Z$, there is a 1-1 correspondence between continuous maps $f: Y\rightarrow Z$ and continuous maps $\tilde{f}:X\rightarrow Z$ such that $\tilde{f}(x)=\tilde{f}(x')$ whenever $(x,x')\in R$. This correspondence is characterized by $\tilde{f}=f\circ \pi$.
And a theorem is used: Let X and Y be topological spaces and let f:X→Y. If X is compact, Y is Hausdorff, and f is a continuous bijection, then f is a homeomorphism between X and Y.
http://mathonline.wikidot.com/homeomorphisms-between-compact-and-hausdorff-spaces
I wonder what is the source of the corollary and theorem, and I wonder if the corollary and theorem have a name.
Your corollary (if I understand you correctly) sounds like „passing to the quotient.“ You could look this up for example in Lee‘s Introduction to Topological Manifols.
Your theorem is a consequence of the „closed map lemma“, which states that a continuous map from a compact space into a Hausdorff space is closed.