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Proving the inverse of a continuous function is also continuous 6 All metric spaces are Hausdorff Given a continuous bijection between a compact space and a Hausdorff space the map is a homeomorphism Proof: We show that f f is a closed map Let K ⊂E1 K ⊂ E 1 be closed then it is compact so f(K) f (K) is compact and compact subsets of Hausdorff spaces are closed Hence, we have that f f is a homeomorphism
Topological properties preserved by continuous maps You'll find topological properties with indication of whether they are preserved by (various kinds of) continuous maps or not (such as open maps, closed maps, quotient maps, perfect maps, etc ) For mere continuous most things have been mentioned: simple covering properties (variations on compactness, connectedness, Lindelöf) and separability