Remark

The interior of a non-empty set can be empty; this is the case for a set consisting of a single point which is not open, for example on the rational line \* (or the real line) *.

Proposition 1 of no. 2 can be restated as follows:

A set is open if and only if it coincides with its interior.

The property $(V_{II})$ of no. 2 implies that every point which is an interior point of each of two subsets $A$ and $B$ is an interior point of $A \cap B$; consequently

Every point which is interior to the complement of a set $A$ is said to be an exterior point of $A$, and the set of these points is called the exterior of $A$ in $X$; a point $x \in X$ which is an exterior point of $A$ is therefore characterized by the property that $x$ has a neighbourhood which does not meet $A$.