Continuous maps from compact to Hausdorff spaces are closed; surjections are quotient maps, injections embeddings, and bijections homeomorphisms.Suggest a slogan
Let \(f\colon X\to Y\) be continuous, with \(X\) compact and \(Y\) Hausdorff. Then
\(f\) is a closed map [@Mun00];
if \(f\) is surjective, \(f\) is a quotient map [@Lee13];
if \(f\) is injective, \(f\) is a topological embedding [@Lee13];
if \(f\) is bijective, \(f\) is a homeomorphism [@Mun00].