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 : 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 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 : X → Y 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).
Let f : X -> Y be a function between topological spaces.
We say that f : X -> Y is an open map if it satisfies any of the following equivalent conditions:
and if B is a basis for X then we may add to this list:
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.
We say that f : X -> Y is a closed map if it satisfies any of the following equivalent conditions:
We say that f : X -> Y is a relatively closed map if f : X -> Im f is a closed map.
The categorical sum of two open maps is open, or of two closed maps is 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.
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 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.
If Y has the discrete topology (i.e. all subsets are open and closed) then every function 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 , the natural projections 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 on the first component; then the set is closed in , but is not closed in . However, for a compact space Y, the projection 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.
Let f : X -> Y be a continuous map that is either open or closed. Then
It is important to remember that Theorem 5.3 says that a function f is continuous if and only if the inverse image of each open set is open. This characterization of continuity should not be confused with another property that a function may or may not possess, the property that the image of each open set is an open set (such functions are called open mappings).
A map F:X -> Y (continuous or not) is said to be an open map if for every closed subset U ? X, F(U) is open in Y, and a closed map if for every closed subset K ? X, F(K) is closed in Y. Continuous maps may be open, closed, both, or neither, as can be seen by examining simple examples involving subsets of the plane.
An open map is a function between two topological spaces which maps open sets to open sets. Likewise, a closed map is a function which maps closed sets to closed sets. The open or closed maps are not necessarily continuous.
Now we are ready for our examples which show that a function may be open without being closed or closed without being open. Also, a function may be simultaneously open and closed or neither open nor closed.(The quoted statement in given in the context of metric spaces but as topological spaces arise as generalizations of metric spaces, the statement holds there as well.)
Exercise 1-19. Show that the projection map ?1:X1 × ··· × Xk -> Xi is an open map, but need not be a closed map. Hint: The projection of R2 onto R is not closed. Similarly, a closed map need not be open since any constant map is closed. For maps that are one-to-one and onto, however, the concepts of 'open' and 'closed' are equivalent.
There are many situations in which a function f:(X,τ)->(Y,τ') has the property that for each open subset A of X, the set f(A) is an open subset of Y, and yet f is not continuous.
Now, the question arises whether the last statement is true in general, that is whether closed maps are continuous. That fails in general as the following example proves.
In general, a map F:X -> Y of a metric space X into a metric space Y may possess any combination of the attributes 'continuous', 'open', and 'closed' (i.e., these are independent concepts).
It seems that the study of open (interior) maps began with papers [13,14] by S. Stoïlow. Clearly, openness of maps was first studied extensively by G.T. Whyburn [19,20].
A composite of open maps is open and a composite of closed maps is closed. Also, a product of open maps is open. In contrast, a product of closed maps is not necessarily closed,...
...let us recall that the composition of open maps is open and the composition of closed maps is closed. Also that the sum of open maps is open and the sum of closed maps is closed. However, the product of closed maps is not necessarily closed, although the product of open maps is open.
Exercise A.32. Suppose are topological spaces. Show that each projection is an open map.