 Open Function
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Open Function

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : XY 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 that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function f : XY is continuous if the preimage of every open set of Y is open in X. (Equivalently, if the preimage of every closed set of Y is closed in X).

Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.

## Definition and characterizations

Let f : X -> Y be a function between topological spaces.

### Open maps

We say that f : X -> Y is an open map if it satisfies any of the following equivalent conditions:

1. f maps open sets to open sets (i.e. for any open subset U of X, f(U) is an open subset of Y);
2. for every x ? X and every neighborhood U of x (however small), there exists a neighborhood V of f (x) such that V ? f (U);
3. f (Int A) ? Int (f (A)) for all subsets A of X, where Int denotes the topological interior of the set;
4. whenever C is a closed subset of X then the set {y ? Y : f -1(y) ? C} is closed in Y;

and if B is a basis for X then we may add to this list:

1. f maps basic open sets to open sets (i.e. for any basic open set B ? B, f (B) is an open subset of Y);

We say that f : X -> Y is a relatively open map if f : X -> Im f is an open map, where Im f is the range or image of f.

Warning: Many authors define "open map" to mean "relatively open map" (e.g. The Encyclopedia of Mathematics). That is, they define "open map" to mean that for any open subset U of X, f (U) is an open subset of Im f (rather than an open subset of Y, which is how this article has defined "open map"). When f is surjective then these two definitions coincide but in general they are not equivalent because although every open map is a relatively open map, relatively open maps often fail to be open maps. It is thus advisable to always check what definition of "open map" an author is using.

### Closed maps

We say that f : X -> Y is a closed map if it satisfies any of the following equivalent conditions:

1. f maps closed sets to closed sets (i.e. for any closed subset U of X, f (U) is an closed subset of Y);
2. ${\overline {f(A)}}\subseteq f({\overline {A}})$ for all subsets A of X.

We say that f : X -> Y is a relatively closed map if f : X -> Im f is a closed map.

## Sufficient conditions

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

The categorical sum of two open maps is open, or of two closed maps is closed.

The categorical product of two open maps is open, however, the categorical 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.

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 the closed map lemma states that if a continuous function between locally compact Hausdorff spaces is proper, then it is also closed.

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.

Invariance of domain — If U is an open subset of Rn and f : U → Rn is an injective continuous map, then V := f(U) is open in Rn and f is a homeomorphism between U and V.

In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map. This theorem has been generalized to topological vector spaces beyond just Banach spaces.

## Examples

Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.

If Y has the discrete topology (i.e. all subsets are open and closed) then every function $f\colon X\to Y$ is both open and closed (but not necessarily continuous). For example, the floor function from R to Z is open and closed, but not continuous. This example shows that the image of a connected space under an open or closed map need not be connected.

Whenever we have a product of topological spaces $X=\prod X_{i}$ , the natural projections $p_{i}\colon X\to X_{i}$ 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. Projections need not be closed however. Consider for instance the projection $p_{1}\colon \mathbf {R} ^{2}\to \mathbf {R}$ on the first component; then the set $A=\{(x,1/x):x\neq 0\}$ is closed in $\mathbf {R} ^{2}$ , but $p_{1}(A)=\mathbf {R} \setminus \{0\}$ is not closed in $\mathbf {R}$ . However, for a compact space Y, the projection $X\times Y\to X$ is closed. This is essentially the tube lemma.

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.

## Properties

Let f : X -> Y be a continuous map that is either open or closed. Then

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.