Open and Closed Maps - Properties

Properties

A function f : X → Y is open if and only if for every x in X and every neighborhood U of x (however small), there exists a neighborhood V of f(x) such that V ⊂ f(U).

It suffices to check openness on a basis for X. That is, a function f : X → Y is open if and only if it maps basic open sets to open sets.

Open and closed maps can also be characterized by the interior and closure operators. Let f : X → Y be a function. Then

• f is open if and only if f(A°) ⊆ f(A)° for all A ⊆ X
• f is closed if and only if f(A)− ⊂ f(A−) for all A ⊂ X

The composition of two open maps is again open; the composition of two closed maps is again closed.

The product of two open maps is open, however the product of two closed maps need not be closed.

A bijective map is open if and only if it is closed. The inverse of a bijective continuous map is a bijective open/closed map (and vice-versa).

A surjective open map is not necessarily a closed map, and likewise a surjective closed map is not necessarily an open map.

Let f : X → Y be a continuous map which is either open or closed. Then

• if f is a surjection, then it is a quotient map,
• if f is an injection, then it is a topological embedding, and
• if f is a bijection, then it is a homeomorphism.

In the first two cases, being open or closed is merely a sufficient condition for the result to follow. In the third case it is necessary as well.