Editing Closed set (section)

== Definition ==
Given a [[topological space]] <math>(X, \tau)</math>, the following statements are equivalent:
# a set <math>A \subseteq X</math> is '''{{em|closed}}''' in <math>X.</math>
# <math>A^c = X \setminus A</math> is an open subset of <math>(X, \tau)</math>; that is, <math>A^{c} \in \tau.</math>
# <math>A</math> is equal to its [[Closure (topology)|closure]] in <math>X.</math>
# <math>A</math> contains all of its [[limit point]]s.
# <math>A</math> contains all of its [[Boundary (topology)|boundary points]]. 

An alternative [[characterization (mathematics)|characterization]] of closed sets is available via [[sequence]]s and [[Net (mathematics)|nets]].  A subset <math>A</math> of a topological space <math>X</math> is closed in <math>X</math> if and only if every [[Limit of a sequence|limit]] of every net of elements of <math>A</math> also belongs to <math>A.</math> In a [[first-countable space]] (such as a metric space), it is enough to consider only convergent [[sequence]]s, instead of all nets.  One value of this characterization is that it may be used as a definition in the context of [[convergence space]]s, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space <math>X,</math> because whether or not a sequence or net converges in <math>X</math> depends on what points are present in <math>X.</math> 
A point <math>x</math> in <math>X</math> is said to be {{em|close to}} a subset <math>A \subseteq X</math> if <math>x \in \operatorname{cl}_X A</math> (or equivalently, if <math>x</math> belongs to the closure of <math>A</math> in the [[topological subspace]] <math>A \cup \{ x \},</math> meaning <math>x \in \operatorname{cl}_{A \cup \{ x \}} A</math> where <math>A \cup \{ x \}</math> is endowed with the [[subspace topology]] induced on it by <math>X</math><ref group="note">In particular, whether or not <math>x</math> is close to <math>A</math> depends only on the [[Topological subspace|subspace]] <math>A \cup \{ x \}</math> and not on the whole surrounding space (e.g. <math>X,</math> or any other space containing <math>A \cup \{ x \}</math> as a topological subspace).</ref>).  
Because the closure of <math>A</math> in <math>X</math> is thus the set of all points in <math>X</math> that are close to <math>A,</math> this terminology allows for a plain English description of closed subsets: 

:a subset is closed if and only if it contains every point that is close to it. 

In terms of net convergence, a point <math>x \in X</math> is close to a subset <math>A</math> if and only if there exists some net (valued) in <math>A</math> that converges to <math>x.</math> 
If <math>X</math> is a [[topological subspace]] of some other topological space <math>Y,</math> in which case <math>Y</math> is called a {{em|topological super-space}} of <math>X,</math> then there {{em|might}} exist some point in <math>Y \setminus X</math> that is close to <math>A</math> (although not an element of <math>X</math>), which is how it is possible for a subset <math>A \subseteq X</math> to be closed in <math>X</math> but to {{em|not}} be closed in the "larger" surrounding super-space <math>Y.</math> 
If <math>A \subseteq X</math> and if <math>Y</math> is {{em|any}} topological super-space of <math>X</math> then <math>A</math> is always a (potentially proper) subset of <math>\operatorname{cl}_Y A,</math> which denotes the closure of <math>A</math> in <math>Y;</math> indeed, even if <math>A</math> is a closed subset of <math>X</math> (which happens if and only if <math>A = \operatorname{cl}_X A</math>), it is nevertheless still possible for <math>A</math> to be a proper subset of <math>\operatorname{cl}_Y A.</math> However, <math>A</math> is a closed subset of <math>X</math> if and only if <math>A = X \cap \operatorname{cl}_Y A</math> for some (or equivalently, for every) topological super-space <math>Y</math> of <math>X.</math> 

Closed sets can also be used to characterize [[Continuous map |continuous functions]]: a map <math>f : X \to Y</math> is [[Continuous function|continuous]] if and only if <math>f\left( \operatorname{cl}_X A \right) \subseteq \operatorname{cl}_Y (f(A))</math> for every subset <math>A \subseteq X</math>; this can be reworded in [[plain English]] as: <math>f</math> is continuous if and only if for every subset <math>A \subseteq X,</math> <math>f</math> maps points that are close to <math>A</math> to points that are close to <math>f(A).</math> Similarly, <math>f</math> is continuous at a fixed given point <math>x \in X</math> if and only if whenever <math>x</math> is close to a subset <math>A \subseteq X,</math> then <math>f(x)</math> is close to <math>f(A).</math>