Compactness is coarsening-preserved

Statement
Suppose $$X$$ is a set and $$\mathcal{T}$$ and $$\mathcal{T}'$$ are two topologies on $$X$$ (in other words, $$(X,\mathcal{T})$$ is a topological space) and $$(X,\mathcal{T}'$$ is a topological space). Further, suppose that $$\mathcal{T}'$$ is a coarser topology than $$\mathcal{T}$$, or equivalently, $$\mathcal{T}$$ is a finer topology than $$\mathcal{T}'$$. In other words, any subset of $$X$$ that is open in $$\mathcal{T}'$$ is open in $$\mathcal{T}$$.

Then, if $$X$$ is compact with topology $$\mathcal{T}$$, it is also compact with topology $$\mathcal{T}'$$.

Stronger facts

 * Weaker than::Compactness is continuous image-closed