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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Suppose is a set and and are two topologies on (in other words, is a topological space) and is a topological space). Further, suppose that is a coarser topology than , or equivalently, is a finer topology than . In other words, any subset of that is open in is open in .
Then, if is compact with topology , it is also compact with topology .