Editing Open and closed maps (section)

==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 {{em|not}} 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 {{em|is}} 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 (topology)|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 '''[[Real number|<math>\R</math>]]''' to '''[[Integer|<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 topology|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<ref>{{cite book|title=General Topology|url=https://archive.org/details/generaltopology00will_0|url-access=registration|first=Stephen|last=Willard|publisher=Addison-Wesley|year=1970|isbn=0486131785}}</ref><ref>{{cite book|last=Lee|first=John M.|date=2012|title=Introduction to Smooth Manifolds|edition=Second|series=Graduate Texts in Mathematics|volume=218|isbn=978-1-4419-9982-5|doi=10.1007/978-1-4419-9982-5|page=606|url=https://zenodo.org/record/4461500|quote='''Exercise A.32.''' Suppose <math>X_1, \ldots, X_k</math> are topological spaces. Show that each projection <math>\pi_i : X_1 \times \cdots \times X_k \to X_i</math> is an open map.}}</ref> (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. 
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 (mathematics)|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.