Template:Random interesting fact

Compact to Hausdorff implies closed (FACT): A continuous map from a compact space to a Hausdorff space is closed.@@@Regularity is not refining-preserved (FACT): Passing to a finer topology does not preserve the property of being a regular space.@@@Compact times paracompact implies paracompact (FACT): This is a cute application of the tube lemma.@@@ Three sides lemma (FACT): The inclusion of three sides of a square, in the filled square, is just like the inclusion of one side.@@@Connected manifold implies homogeneous (FACT): There is a homeomorphism taking any point of a connected manifold to any other point.