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 <math>f : X \to Y</math> is open if for any open set <math>U</math> in <math>X,</math> the image <math>f(U)</math> is open in <math>Y.</math>

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.

Although their definitions seem more natural, open and closed maps are much less important than continuous maps.

Recall that, by definition, a function <math>f : X \to Y</math> is continuous if and only if the preimage of every open set of <math>Y</math> is open in <math>X.</math>

Definitions and characterizations

If <math>S</math> is a subset of a topological space then let <math>\overline{S}</math> and <math>\operatorname{Cl} S</math> (resp. <math>\operatorname{Int} S</math>) denote the closure (resp. interior) of <math>S</math> in that space.

Let <math>f : X \to Y</math> be a function between topological spaces. If <math>S</math> is any set then <math>f(S) := \left\{ f(s) ~:~ s \in S \cap \operatorname{domain} f \right\}</math> is called the image of <math>S</math> under <math>f.</math>

Competing definitions

There are two different competing, but closely related, definitions of "" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets."

The following terminology is sometimes used to distinguish between the two definitions.

A map <math>f : X \to Y</math> is called a

  • "" if whenever <math>U</math> is an open subset of the domain <math>X</math> then <math>f(U)</math> is an open subset of <math>f</math>'s codomain <math>Y.</math>
  • "" if whenever <math>U</math> is an open subset of the domain <math>X</math> then <math>f(U)</math> is an open subset of <math>f</math>'s image <math>\operatorname{Im} f := f(X),</math> where as usual, this set is endowed with the subspace topology induced on it by <math>f</math>'s codomain <math>Y.</math>

Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.

:Warning: Many authors define "open map" to mean " open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean " open map". In general, these definitions are equivalent so it is thus advisable to always check what definition of "open map" an author is using.

A surjective map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.

More generally, a map <math>f : X \to Y</math> is relatively open if and only if the surjection <math>f : X \to f(X)</math> is a strongly open map.

Because <math>X</math> is always an open subset of <math>X,</math> the image <math>f(X) = \operatorname{Im} f</math> of a strongly open map <math>f : X \to Y</math> must be an open subset of its codomain <math>Y.</math> In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain.

In summary,

:A map is strongly open if and only if it is relatively open and its image is an open subset of its codomain.

By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition.

The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".

Open maps

A map <math>f : X \to Y</math> is called an or a if it satisfies any of the following equivalent conditions:

<ol>

<li>Definition: <math>f : X \to Y</math> maps open subsets of its domain to open subsets of its codomain; that is, for any open subset <math>U</math> of <math>X</math>, <math>f(U)</math> is an open subset of <math>Y.</math></li>

<li><math>f : X \to Y</math> is a relatively open map and its image <math>\operatorname{Im} f := f(X)</math> is an open subset of its codomain <math>Y.</math></li>

<li>For every <math>x \in X</math> and every neighborhood <math>N</math> of <math>x</math> (however small), <math>f(N)</math> is a neighborhood of <math>f(x)</math>. We can replace the first or both instances of the word "neighborhood" with "open neighborhood" in this condition and the result will still be an equivalent condition:

  • For every <math>x \in X</math> and every open neighborhood <math>N</math> of <math>x</math>, <math>f(N)</math> is a neighborhood of <math>f(x)</math>.
  • For every <math>x \in X</math> and every open neighborhood <math>N</math> of <math>x</math>, <math>f(N)</math> is an open neighborhood of <math>f(x)</math>.</li>

<li><math>f\left( \operatorname{Int}_X A \right) \subseteq \operatorname{Int}_Y ( f(A) )</math> for all subsets <math>A</math> of <math>X,</math> where <math>\operatorname{Int}</math> denotes the topological interior of the set.</li>

<li>Whenever <math>C</math> is a closed subset of <math>X</math> then the set <math>\left\{ y \in Y ~:~ f^{-1}(y) \subseteq C \right\}</math> is a closed subset of <math>Y.</math>

  • This is a consequence of the identity <math>f(X \setminus R) = Y \setminus \left\{ y \in Y : f^{-1}(y) \subseteq R \right\},</math> which holds for all subsets <math>R \subseteq X.</math></li>

</ol>

If <math>\mathcal{B}</math> is a basis for <math>X</math> then the following can be appended to this list:

  1. <li value="6"><math>f</math> maps basic open sets to open sets in its codomain (that is, for any basic open set <math>B \in \mathcal{B},</math> <math>f(B)</math> is an open subset of <math>Y</math>).</li>

Closed maps

A map <math>f : X \to Y</math> is called a if whenever <math>C</math> is a closed subset of the domain <math>X</math> then <math>f(C)</math> is a closed subset of <math>f</math>'s image <math>\operatorname{Im} f := f(X),</math> where as usual, this set is endowed with the subspace topology induced on it by <math>f</math>'s codomain <math>Y.</math>

A map <math>f : X \to Y</math> is called a or a if it satisfies any of the following equivalent conditions:

<ol>

<li>Definition: <math>f : X \to Y</math> maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset <math>C</math> of <math>X,</math> <math>f(C)</math> is a closed subset of <math>Y.</math>

<li><math>f : X \to Y</math> is a relatively closed map and its image <math>\operatorname{Im} f := f(X)</math> is a closed subset of its codomain <math>Y.</math></li>

<li><math>\overline{f(A)} \subseteq f\left(\overline{A}\right)</math> for every subset <math>A \subseteq X.</math></li>

<li><math>\overline{f(C)} \subseteq f(C)</math> for every closed subset <math>C \subseteq X.</math></li>

<li>Whenever <math>U</math> is an open subset of <math>X</math> then the set <math>\left\{y \in Y ~:~ f^{-1}(y) \subseteq U\right\}</math> is an open subset of <math>Y.</math></li>

<li>If <math>x_{\bull}</math> is a net in <math>X</math> and <math>y \in Y</math> is a point such that <math>f\left(x_{\bull}\right) \to y</math> in <math>Y,</math> then <math>x_{\bull}</math> converges in <math>X</math> to the set <math>f^{-1}(y).</math>

  • The convergence <math>x_{\bull} \to f^{-1}(y)</math> means that every open subset of <math>X</math> that contains <math>f^{-1}(y)</math> will contain <math>x_j</math> for all sufficiently large indices <math>j.</math></li>

</ol>

A surjective map is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent.

By definition, the map <math>f : X \to Y</math> is a relatively closed map if and only if the surjection <math>f : X \to \operatorname{Im} f</math> is a strongly closed map.

If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") is to continuity.

This does not happen with the definition of "open map" (which is: "every image of an open set is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in general equivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any set <math>S,</math> only <math>f(X \setminus S) \supseteq f(X) \setminus f(S)</math> is guaranteed in general, whereas for preimages, equality <math>f^{-1}(Y \setminus S) = f^{-1}(Y) \setminus f^{-1}(S)</math> always holds.

Examples

The function <math>f : \R \to \R</math> defined by <math>f(x) = x^2</math> is continuous, closed, and relatively open, but not (strongly) open. This is because if <math>U = (a, b)</math> is any open interval in <math>f</math>'s domain <math>\R</math> that does contain <math>0</math> then <math>f(U) = (\min \{ a^2, b^2 \}, \max \{ a^2, b^2 \}),</math> where this open interval is an open subset of both <math>\R</math> and <math>\operatorname{Im} f := f(\R) = [0, \infty).</math> However, if <math>U = (a, b)</math> is any open interval in <math>\R</math> that contains <math>0</math> then <math>f(U) = [0, \max \{ a^2, b^2 \}),</math> which is not an open subset of <math>f</math>'s codomain <math>\R</math> but an open subset of <math>\operatorname{Im} f = [0, \infty).</math> Because the set of all open intervals in <math>\R</math> is a basis for the Euclidean topology on <math>\R,</math> this shows that <math>f : \R \to \R</math> is relatively open but not (strongly) open.

If <math>Y</math> has the discrete topology (that is, all subsets are open and closed) then every function <math>f : X \to Y</math> is both open and closed (but not necessarily continuous).

For example, the floor function from <math>\R</math> to <math>\Z</math> 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 <math display="inline">X=\prod X_i,</math> the natural projections <math>p_i : X \to X_i</math> 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 <math>p_1 : \R^2 \to \R</math> on the first component; then the set <math>A = \{(x, 1/x) : x \neq 0\}</math> is closed in <math>\R^2,</math> but <math>p_1(A) = \R \setminus \{0\}</math> is not closed in <math>\R.</math>

However, for a compact space <math>Y,</math> the projection <math>X \times Y \to X</math> is closed. This is essentially the tube lemma.

To every point on the unit circle we can associate the angle of the positive <math>x</math>-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval <nowiki>[0,2&pi;)</nowiki> 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.

Sufficient conditions

Every homeomorphism is open, closed, and continuous. In fact, a continuous bijection is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.

The composition of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map. However, the composition of two relatively open maps need not be relatively open and the composition of two relatively closed maps need not be relatively closed.

If <math>f : X \to Y</math> is strongly open (respectively, strongly closed) and <math>g : Y \to Z</math> is relatively open (respectively, relatively closed), then <math>g \circ f : X \to Z</math> is relatively open (respectively, relatively closed).

Let <math>f : X \to Y</math> be a map. Given any subset <math>T \subseteq Y</math>, if <math>f</math> is relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, surjective), then the same is true of its restriction

<math display=block>f\big\vert_{f^{-1}(T)} ~:~ f^{-1}(T) \to T</math>

to the <math>f</math>-saturated subset <math>f^{-1}(T)</math>.

The categorical sum of two open maps is open, and of two closed maps is closed.</li>

</ul>

See also

Notes

Citations

References

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