Editing Interior (topology) (section)

==Definitions==

===Interior point===

If <math>S</math> is a subset of a [[Euclidean space]], then <math>x</math> is an interior point of <math>S</math> if there exists an [[open ball]] centered at <math>x</math> which is completely contained in <math>S.</math>
(This is illustrated in the introductory section to this article.)

This definition generalizes to any subset <math>S</math> of a [[metric space]] <math>X</math> with metric <math>d</math>: <math>x</math> is an interior point of <math>S</math> if there exists a real  number <math>r > 0,</math> such that <math>y</math> is in <math>S</math> whenever the distance <math>d(x, y) < r.</math>

This definition generalizes to [[topological space]]s by replacing "open ball" with "[[open set]]".
If <math>S</math> is a subset of a topological space <math>X</math> then <math>x</math> is an {{em|interior point}} of <math>S</math> in <math>X</math> if <math>x</math> is contained in an open subset of <math>X</math> that is completely contained in <math>S.</math>
(Equivalently, <math>x</math> is an interior point of <math>S</math> if <math>S</math> is a [[Neighbourhood (mathematics)|neighbourhood]] of <math>x.</math>)

===Interior of a set===

The '''interior''' of a subset <math>S</math> of a topological space <math>X,</math> denoted by <math>\operatorname{int}_X S</math> or <math>\operatorname{int} S</math> or <math>S^\circ,</math> can be defined in any of the following equivalent ways:
# <math>\operatorname{int} S</math> is the largest open subset of <math>X</math> contained in <math>S.</math>
# <math>\operatorname{int} S</math> is the union of all open sets of <math>X</math> contained in <math>S.</math>
# <math>\operatorname{int} S</math> is the set of all interior points of <math>S.</math>
If the space <math>X</math> is understood from context then the shorter notation <math>\operatorname{int} S</math> is usually preferred to <math>\operatorname{int}_X S.</math>