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Note that neither open nor closed maps are required to be 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''=Π''X''''i'', then the natural projections ''p''''i'' : ''X'' → ''X''''i'' are open (as well as continuous). Since the projections of Fiber Bundle s and Covering Map s are locally natural projections of products, these are also open maps. (Note that product projections need not be closed. Consider for instance the projection ''p''1 : R2 → R on the first component; ''A'' = {(''x'',1/''x'') : ''x''≠0} is closed in R2, but ''p''1(''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 The function ''f'' : R → R with ''f''(''x'') = ''x''2 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 . 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
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 Space s 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. |
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