In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.[1][2][3] That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets.[3][4] A map may be open, closed, both, or neither;[5] in particular, an open map need not be closed and vice versa.[6]
Open[7] and closed[8] maps are not necessarily continuous.[4] Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;[3] this fact remains true even if one restricts oneself to metric spaces.[9] Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function is continuous if the preimage of every open set of is open in [2] (Equivalently, if the preimage of every closed set of is closed in ).
Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.[10]
It is important to remember that Theorem 5.3 says that a function 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 (continuous or not) is said to be an open map if for every closed subset is open in and a closed map if for every closed subset is closed in 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 π1:X1 × ··· × Xk → Xi is an open map, but need not be a closed map. Hint: The projection of R2 onto 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 has the property that for each open subset of the set is an open subset of and yet 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 of a metric space into a metric space may possess any combination of the attributes 'continuous', 'open', and 'closed' (that is, 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].