Editing Interior (topology) (section)

==Examples==
[[File:Set of real numbers with epsilon-neighbourhood.svg|thumb|<math>a</math> is an interior point of <math>M</math> because there is an ε-neighbourhood of <math>a</math> which is a subset of <math>M.</math>]]

*In any space, the interior of the [[empty set]] is the empty set.
*In any space <math>X,</math> if <math>S \subseteq X,</math> then <math>\operatorname{int} S \subseteq S.</math>
*If <math>X</math> is the [[real line]] <math>\Reals</math> (with the standard topology), then <math>\operatorname{int} ([0, 1]) = (0, 1)</math> whereas the interior of the set <math>\Q</math> of [[rational number]]s is empty: <math>\operatorname{int} \Q = \varnothing.</math>
*If <math>X</math> is the [[Complex number|complex plane]] <math>\Complex,</math> then <math>\operatorname{int} (\{z \in \Complex : |z| \leq 1\}) = \{z \in \Complex : |z| < 1\}.</math>
*In any [[Euclidean space]], the interior of any [[finite set]] is the empty set.

On the set of [[real number]]s, one can put other topologies rather than the standard one:

*If <math>X</math> is the real numbers <math>\Reals</math> with the [[lower limit topology]], then <math>\operatorname{int} ([0, 1]) = [0, 1).</math>
*If one considers on <math>\Reals</math> the topology in which [[Discrete topology|every set is open]], then <math>\operatorname{int} ([0, 1]) = [0, 1].</math>
*If one considers on <math>\Reals</math> the topology in which the only open sets are the empty set and <math>\Reals</math> itself, then <math>\operatorname{int} ([0, 1])</math> is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space.
The last two examples are special cases of the following.

*In any [[discrete space]], since every set is open, every set is equal to its interior.
*In any [[indiscrete space]] <math>X,</math> since the only open sets are the empty set and <math>X</math> itself, <math>\operatorname{int} X = X</math> and for every [[subset|proper subset]] <math>S</math> of <math>X,</math> <math>\operatorname{int} S</math> is the empty set.