# Compactness is coarsening-preserved

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 $\mathcal{T}'$.