= Compact Hausdorff reflection
{title2=$R:\mathbf{Top}\to\mathbf{KHaus}$}
The inclusion of <compact Hausdorff spaces> in <topological spaces> has a <left adjoint>. Its unit $X\to RX$ is universal among <continuous functions> from $X$ to compact Hausdorff spaces and has <dense> image. Completeness follows from <products in a category> and closed <equalizers>. For any map $X\to K$, its closed image has a dense subset of cardinality at most $|X|$ and hence cardinality at most $2^{2^{|X|}}$. Representatives of all such compact Hausdorff spaces and maps establish the <solution-set condition>, so the <general adjoint functor theorem> gives the reflection. The unit need not be injective or a topological embedding for an arbitrary $X$.
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