Compactness is coarsening-preserved

This article gives the statement, and possibly proof, of a topological space property (i.e., compact space) satisfying a topological space metaproperty (i.e., coarsening-preserved property of topological spaces)
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Statement

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

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

Related facts

Stronger facts

• Compactness is continuous image-closed