Open and closed maps
In topology, an open map is a function between two topological spaces which maps open sets to open sets.[1][2][3] 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.[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 f : X → Y is continuous if the preimage of every open set of Y is open in X.[2] (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.[10]
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 : X → 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=ΠXi, the natural projections pi : X → Xi are open[11][12] (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 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. However, for compact Y, the projection X × Y → 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
A function f : X → Y is open if and only if for every x in X and every neighborhood U of x (however small), there exists a neighborhood V of f(x) such that V ⊂ f(U).
It suffices to check openness on a basis for X. That is, a function f : X → Y is open if and only if it maps basic open sets to open sets.
Open and closed maps can also be characterized by the interior and closure operators. Let f : X → Y be a function. Then
- f is open if and only if f(A°) ⊆ f(A)° for all A ⊆ X
- f is closed if and only if f(A)− ⊂ f(A−) for all A ⊂ X
The composition of two open maps is again open; the composition of two closed maps is again closed.[13][14]
The categorical sum of two open maps is open, or of two closed maps is closed.[14]
The categorical product of two open maps is open, however the categorical product of two closed maps need not be closed.[13][14]
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.
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.
Open and closed mapping theorems
It is useful to have conditions for determining when a map is open or closed. The following are some results along these lines.
The closed map lemma states that 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.
See also
References
- ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
- ^ a b Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 89. ISBN 0-486-66352-3.
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 b c Lee, John M. (2003). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218. Springer Science & Business Media. p. 550. ISBN 9780387954486.
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.
- ^ a b Ludu, Andrei. Nonlinear Waves and Solitons on Contours and Closed Surfaces. Springer Series in Synergetics. p. 15. ISBN 9783642228940.
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.
- ^ Sohrab, Houshang H. (2003). Basic Real Analysis. Springer Science & Business Media. p. 203. ISBN 9780817642112.
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.) - ^ Naber, Gregory L. (2012). Topological Methods in Euclidean Spaces. Dover Books on Mathematics (reprint ed.). Courier Corporation. p. 18. ISBN 9780486153445.
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 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.
- ^ Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 89. ISBN 0-486-66352-3.
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.
- ^ Boos, Johann (2000). Classical and Modern Methods in Summability. Oxford University Press. p. 332. ISBN 0-19-850165-X.
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.
- ^ Kubrusly, Carlos S. (2011). The Elements of Operator Theory. Springer Science & Business Media. p. 115. ISBN 9780817649982.
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).
- ^ Hart, K. P.; Nagata, J.; Vaughan, J. E., eds. (2004). Encyclopedia of General Topology. Elsevier. p. 86. ISBN 0-444-50355-2.
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].
- ^ Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 0486131785.
- ^ Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). p. 606. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9982-5.
Exercise A.32. Suppose X1,...,Xk are topological spaces. Show that each projection πi: X1 × ··· × Xk → Xi is an open map.
- ^ a b Baues, Hans-Joachim; Quintero, Antonio (2001). Infinite Homotopy Theory. K-Monographs in Mathematics. Vol. 6. p. 53. ISBN 9780792369820.
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,...
- ^ a b c James, I. M. (1984). General Topology and Homotopy Theory. Springer-Verlag. p. 49. ISBN 9781461382836.
...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.