Open and closed maps

In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X → Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets.

Note that neither open nor closed maps are required to be continuous. Although their definitions seem natural, open and closed maps are much less important than continuous maps. Recall that a function f : X → Y is continuous if the preimage of any open set of Y is open in X, or equivalently: if the preimage of every closed set of Y is closed in X.

Examples

Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism iff it's open, or equivalently, iff it's closed.

If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous).

Whenever we have a product of topological spaces X=ΠX_i, then the natural projections p_i : X → X_i are open (as well as continuous). Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps. (Note that product projections need not be closed. Consider for instance the projection p₁ : R² → R on the first component; A = {(x,1/x) : x≠0} is closed in R², but p₁(A) = R-{0} is not closed.)

To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying the codomain is essential!

The function f : R → R with f(x) = x² is continuous and closed, but not open.

The floor function from R to Z is open and closed (because Z carries the discrete topology). This example shows that the image of a connected space under an open or closed map need not be connected.

Facts and theorems

A function f : X → Y is open iff

to every x in X and to every neighborhood U of x (however small), there exists a neighborhood V of f(x) such that <math>V \subseteq f(U)<math>.

A function f : X → Y is closed iff

whenever (x_α) is a net in X such that (f(x_α)) has limit y, then (x_α) has a subnet that converges towards a preimage of y.

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

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

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.

A very useful result regarding closed maps is the closed map lemma: every continuous function f : X → Y from a compact space X to a Hausdorff space Y is closed and proper (i.e. preimages of compact sets are compact). A variant of this result states that if a continuous function between locally compact Hausdorff spaces is proper, then it is also closed.

In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map.

In complex analysis, the identically named open mapping theorem states that every non-constant holomorphic function defined on a connected open subset of the complex plane is an open map.

The invariance of domain theorem states that a continuous and locally injective function between two n-dimensional topological manifolds must be open.pl:Odwzorowanie otwarte