TITLE: Ambiguity in definition of compactness
QUESTION [7 upvotes]: I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes:

A topological space $X$ is compact if every open cover has a finite
  subcover on $X$.

Okay, this seems to make sense. But, an example is later presented in the notes: $X$ with the trivial topology $\tau = \{X, \varnothing \}$ is compact. Again, okay, this seems to make sense, as any open cover is finite, if we look at $\tau$. 
My question is: Does the cover come from the set $X$ or $\tau$? Apologies if my thought process seems unclear!

REPLY [9 votes]: open cover (for a given topology $\tau$): a collection of open sets $U_{\alpha}$ (open with respect to the topology $\tau$) such that $$X \subset \underset{\alpha \in A}{\bigcup} U_{\alpha}$$
open set: A set in the topology $\tau$ of the space $X$.
compact (for a given topology $\tau$): every open cover of $X$ (with respect to the topology $\tau$) has a finite subcover.
A given open cover therefore depends on both:

the given space $X$
the topology $\tau$ given for $X$

If $\nu$ is a different topology for $X$, then $(X,\tau)$ and $(X,\nu)$ are different topological spaces. Therefore, which collections of sets are open covers of $X$ and as a result whether or not $X$ is compact will be different.