I was reading a proof online and it linked to a book by Munkres which says
Every closed subspace of a compact space is compact.
I dug out the book and searched the index for this term. Unfortunately it was not clearly defined, so I am assuming it just means its a subspace with closed sets in the topology? Because that's what the proof seems to imply and the word "closed subspace" conflicts with what I used to remember in linear algebra.
Best Answer
A closed subspace is a subspace that when treated as a subset of the original space is a closed set in the original topology.
That is, if $(X,\tau)$ is a topological space, then $Y \subseteq X$ is a closed subspace of $X$ (When equipped with the subspace topology) when $X \setminus Y \in \tau$.
Similarly, if $Y \in \tau$, then $Y$ is an open subspace of $X$ when equipped with the subspace topology.