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 ${\mathcal {T}}'$ 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 ${\mathcal {T}}'$.

Related facts

Stronger facts

• Compactness is continuous image-closed