How to prove compactness is a topological property by definition?

Question

I learned from the lectures that a property of a set which remains invariant under a homeomorphism is called topological property. And $S$ is compact in $X$ if every open cover of $S$ has a finite sub cover. But I do not know how to connect these two definitions to prove that compactness is a topological property.

Answer 1

The wry term topological mapping most likely means a homeomorphism,
a continuous bijection with a continuous inverse.
Topological properties are about spaces.
They are not about subsets.

Let $X$ be a compact space and $h:X\to Y$ a homeomorphism.
Let $C$ be an open cover of $Y$.
Show $\{ h^{-1}(A) : A \in C\}$ is an open cover of $X$.
Is this a sufficient start of a proof to encourage you to finish it?