Compactness is coarsening-preserved

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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}'.

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