Closure, Interior & Limit Points

Closure adds touched points, interior keeps surrounded ones, limit points are approached.

The idea

The interior, closure, and limit points of a set describe how the set sits inside its space: which points the set surrounds, and which points it comes arbitrarily close to. They express nearness in any topological space, with no distance available.

Take a subset $A$ of a topological space. A point $x$ lies in the interior $\operatorname{int} A$ when some open set contains $x$ and is itself contained in $A$; the interior collects the points $A$ surrounds with room to spare, and it is the largest open set contained in $A$. A point $x$ is a limit point of $A$ when every open set containing $x$ meets $A$ at a point other than $x$ itself; the limit points are the points $A$ approaches arbitrarily closely, whether or not it contains them. The closure $\overline{A}$ is $A$ together with all its limit points, and it is the smallest set containing $A$ whose complement is open.

The clause "other than $x$ itself" separates a point $A$ approaches from a point $A$ merely contains. A point of $A$ with an open set around it meeting $A$ nowhere else is isolated: it belongs to the closure, but only because it belongs to $A$. The points where the two operators disagree form the boundary $\overline{A} \setminus \operatorname{int} A$, the points that $A$ and its complement both touch.

Ways to work on it

• Walkthrough. Interior, closure, limit points, and boundary of a sample set.
• Practice. Identify the closure of a subset of the real line.
• Hardest. Interior and closure of the rationals in an interval.

Not sure where to start? Take the ten-question placement test.