Open and closed maps

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 $f:X\to Y$ 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.[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 $f:X\to Y$ 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]

Definition and characterizations

Let $f:X\to Y$ be a function between topological spaces.

Open maps

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

$f$ maps open sets to open sets (i.e. for any open subset $U$ of $X,$ $f(U)$ is an open subset of $Y$ );
for every $x\in X$ and every neighborhood $U$ of $x$ (however small), there exists a neighborhood $V$ of $f(x)$ such that $V\subseteq f(U)$ ;
$f\left(\operatorname {Int} A\right)\subseteq \operatorname {Int} \left(f(A)\right)$ for all subsets $A$ of $X,$ where $\operatorname {Int}$ denotes the topological interior of the set;
whenever $C$ is a closed subset of $X$ then the set $\left\{y\in Y:f^{-1}(y)\subseteq C\right\}$ is closed in $Y$ ;[11]

and if ${\mathcal {B}}$ is a basis for $X$ then we may add to this list:

$f$ maps basic open sets to open sets (i.e. for any basic open set $B\in {\mathcal {B}},$ $f(B)$ is an open subset of $Y$ );

We say that $f:X\to Y$ is a relatively open map if $f:X\to \operatorname {Im} f$ is an open map, where $\operatorname {Im} f$ is the range or image of $f.$ [12]

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

\operatorname {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 $f:X\to Y$ is a closed map if it satisfies any of the following equivalent conditions:

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

We say that $f:X\to Y$ is a relatively closed map if $f:X\to \operatorname {Im} f$ is a closed map.

Sufficient conditions

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

Closed map lemma — Every continuous function $f:X\to Y$ 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 $\mathbb {R} ^{n}$ and $f:U\to \mathbb {R} ^{n}$ is an injective continuous map, then $V:=f(U)$ is open in $\mathbb {R} ^{n}$ 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 $f:X\to Y$ is both open and closed (but not necessarily continuous). For example, the floor function from $\mathbb {R}$ to $\mathbb {Z}$ 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 $p_{i}:X\to X_{i}$ are open[15][16] (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 $p_{1}:\mathbf {R} ^{2}\to \mathbf {R}$ on the first component; then the set $A=\{\left(x,1/x\right):x\neq 0\}$ 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 $f:\mathbb {R} \to \mathbb {R}$ with $f(x)=x^{2}$ is continuous and closed, but not open.

Properties

Let $f:X\to Y$ be a continuous map that 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.

If $f:X\to Y$ is a continuous open map, $A\subseteq X,$ and $S\subseteq Y,$ then:

$f^{-1}\left(\operatorname {Bd} _{Y}S\right)=\operatorname {Bd} _{X}\left(f^{-1}(S)\right)$ where $\operatorname {Bd}$ denotes the topological boundary of a set.
$f^{-1}\left({\overline {S}}\right)={\overline {f^{-1}(S)}}$ where ${\overline {S}}$ denote the topological closure of a set.
If ${\overline {A}}={\overline {\operatorname {Int} A}}$ then
${\overline {\operatorname {Int} _{Y}f(A)}}={\overline {f(A)}}={\overline {f\left(\operatorname {Int} _{X}A\right)}}={\overline {f\left({\overline {\operatorname {Int} _{X}A}}\right)}}$

where this set is also a regular closed set.[note 1] In particular, if $A$ is a regular closed set then so is ${\overline {f(A)}}.$ And if $A$ a regular open set then so is $Y\setminus {\overline {f(X\setminus A)}}.$
If the continuous open map $f:X\to Y$ is also surjective then $\operatorname {Int} _{X}f^{-1}(S)=f^{-1}\left(\operatorname {Int} _{Y}S\right)$ and moreover, $S$ is a regular open (resp. a regular closed)[note 1] subset of $Y$ if and only if $f^{-1}(S)$ is a regular open (resp. a regular closed) subset of $X.$

Suppose $F:X\to Z$ is a function and $\pi :X\to Y$ is a surjective map. Let $D:=\left\{y\in Y~:~F{\text{ is constant on the fiber }}\pi ^{-1}(y)\right\}$ so that it for every $d\in D,$ $F\left(\pi ^{-1}(d)\right)$ is a singleton set whose sole element will be denoted by $f(d).$ This induces a map $f:D\to Z,$ which is the unique map satisfying $F\left(\pi ^{-1}(d)\right)=\{f(d)\}$ for every $d\in D.$ The importance of this map $f$ is that $F=f\circ \pi$ holds on $\pi ^{-1}(D)$ where by its very definition, the set $D$ is the (unique) largest subset of $Y$ for which this is true. If $\pi :X\to Y$ is a continuous open surjection from a first-countable space $X$ onto a Hausdorff space $Y,$ and if $F:X\to Z$ is a continuous map valued in a Hausdorff space $Z,$ then $f:D\to Z$ is continuous[note 2] and $D$ is a closed subset of $Y.$ [note 3]

Notes

A subset $S\subseteq X$ is called a regular closed set if ${\overline {\operatorname {Int} S}}=S$ or equivalently, if $\operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S,$ where $\operatorname {Bd} S$ (resp. $\operatorname {Int} S,$ ${\overline {S}}$ ) denotes the topological boundary (resp. interior, closure) of $S$ in $X.$ The set $S$ is called a regular open set if $\operatorname {Int} \left({\overline {S}}\right)=S$ or equivalently, if $\operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S.$ The interior (taken in $X$ ) of a closed subset of $X$ is always a regular open subset of $X.$ The closure (taken in $X$ ) of an open subset of $X$ is always a regular closed subset of $X.$
This is a consequence of the following more general result, which is straightforward to prove: If $\pi :X\to Y$ is a quotient map and $F:X\to Z$ is arbitrary then with $D$ and $f:D\to Z$ defined as before, for any $A\subseteq D,$ the restriction $F{\big \vert }_{\pi ^{-1}(A)}:\pi ^{-1}(A)\to Z$ is continuous if and only $f{\big \vert }_{R}:R\to Z$ is continuous.
The (non-trivial) conclusion that $D$ is a closed subset of $Y$ was reached despite the fact that the definition of $D$ was purely set-theoretic and completely independent of any topology.

Citations

Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
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).
Lee, John M. (2003). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218. Springer Science & Business Media. p. 550. ISBN 9780387954486. A map $F:X\to Y$ (continuous or not) is said to be an open map if for every closed subset $U\subseteq X,$ $F(U)$ is open in $Y,$ and a closed map if for every closed subset $K\subseteq U,$ $F(K)$ 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.
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 $\pi _{i}:X_{i}\times \cdots \times X_{k}\to X_{i}$ π₁:X₁ × ··· × X_k → X_i is an open map, but need not be a closed map. Hint: The projection of R² onto $\mathbb {R}$ 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 $f:\left(X,\tau \right)\to \left(Y,\tau '\right)$ 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 $F:X\to Y$ 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].
Stack exchange post
Narici & Beckenstein 2011, pp. 225-273.
Baues, Hans-Joachim; Quintero, Antonio (2001). Infinite Homotopy Theory. K-Monographs in Mathematics. 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,...
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.
Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 0486131785.
Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218 (Second ed.). p. 606. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9982-5. Exercise A.32. Suppose $X_{1},\ldots ,X_{k}$ are topological spaces. Show that each projection $\pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}$ is an open map.

References

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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[DefOfRegularOpenClosed-17] A subset $S\subseteq X$ is called a regular closed set if ${\overline {\operatorname {Int} S}}=S$ or equivalently, if $\operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S,$ where $\operatorname {Bd} S$ (resp. $\operatorname {Int} S,$ ${\overline {S}}$ ) denotes the topological boundary (resp. interior, closure) of $S$ in $X.$ The set $S$ is called a regular open set if $\operatorname {Int} \left({\overline {S}}\right)=S$ or equivalently, if $\operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S.$ The interior (taken in $X$ ) of a closed subset of $X$ is always a regular open subset of $X.$ The closure (taken in $X$ ) of an open subset of $X$ is always a regular closed subset of $X.$

[CharacterizationOfContinuity-18] This is a consequence of the following more general result, which is straightforward to prove: If $\pi :X\to Y$ is a quotient map and $F:X\to Z$ is arbitrary then with $D$ and $f:D\to Z$ defined as before, for any $A\subseteq D,$ the restriction $F{\big \vert }_{\pi ^{-1}(A)}:\pi ^{-1}(A)\to Z$ is continuous if and only $f{\big \vert }_{R}:R\to Z$ is continuous.

[19] The (non-trivial) conclusion that $D$ is a closed subset of $Y$ was reached despite the fact that the definition of $D$ was purely set-theoretic and completely independent of any topology.

[1] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

[mendelson-2] 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).

[lee550-3] Lee, John M. (2003). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218. Springer Science & Business Media. p. 550. ISBN 9780387954486. A map $F:X\to Y$ (continuous or not) is said to be an open map if for every closed subset $U\subseteq X,$ $F(U)$ is open in $Y,$ and a closed map if for every closed subset $K\subseteq U,$ $F(K)$ 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.

[ludu15-4] 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.

[5] 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.)

[6] 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 $\pi _{i}:X_{i}\times \cdots \times X_{k}\to X_{i}$ π₁:X₁ × ··· × X_k → X_i is an open map, but need not be a closed map. Hint: The projection of R² onto $\mathbb {R}$ 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.

[mendelson2-7] 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 $f:\left(X,\tau \right)\to \left(Y,\tau '\right)$ 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.

[8] 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.

[9] Kubrusly, Carlos S. (2011). The Elements of Operator Theory. Springer Science & Business Media. p. 115. ISBN 9780817649982. In general, a map $F:X\to Y$ 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).

[10] 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].

[11] Stack exchange post

[FOOTNOTENariciBeckenstein2011225-273-12] Narici & Beckenstein 2011, pp. 225-273.

[baues55-13] Baues, Hans-Joachim; Quintero, Antonio (2001). Infinite Homotopy Theory. K-Monographs in Mathematics. 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,...

[james49-14] 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.

[15] Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 0486131785.

[16] Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218 (Second ed.). p. 606. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9982-5. Exercise A.32. Suppose $X_{1},\ldots ,X_{k}$ are topological spaces. Show that each projection $\pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}$ is an open map.