A closed map sends closed subsets of its domain to closed subsets of its codomain. A bijection is a homeomorphism exactly when it is continuous and closed.

Compact-preimage theorem for closed maps

Let

f : X \to Y

be a continuous map that is a closed map and whose fibres are compact. Then

f^{- 1} (K)

K \subseteq Y

, without a Hausdorff hypothesis. Given an open cover, cover each fibre by a finite union

V_{y}

of its members. The set

W_{y} = Y ∖ f (X ∖ V_{y})

is open, contains

y

, and has

f^{- 1} (W_{y}) \subseteq V_{y}

. A finite subcover of

K

by the

W_{y}

uses only finitely many original cover members and covers

f^{- 1} (K)

. Empty fibres cause no exception.

 Ancestors (5)

There are no discussions about this article yet.

Homeomorphism is a concept in topology, a branch of mathematics that studies the properties of space that are preserved under continuous transformations. Specifically, a homeomorphism is a continuous function between two topological spaces that has a continuous inverse. Formally, let \( X \) and \( Y \) be topological spaces.