The Interior Points of Sets in a Topological Space

# The Interior Points of Sets in a Topological Space

Recall from the The Open Neighbourhoods of Points in a Topological Space page that if $(X, \tau)$ is a topological space and $x \in X$ then an open neighbourhood of $x$ is any open set $U$ ($U \in \tau$) such that $x \in U$.

Given a subset $A \subseteq X$, we will give a special name to the points $a \in A$ that contain an open neighbourhood $U$ fully contained in $A$.

 Definition: Let $(X, \tau)$ be a topological space and let $A \subseteq X$. A point $a \in A$ is called an Interior Point of $A$ if there exists an open neighbourhood $U$ ($U \in \tau$) of $a$ such that $a \in U \subseteq A$. The set of all interior points of $A$ is called the Interior of $A$ and is denoted $\mathrm{int} (A)$.

Let's now look at some simple results regarding interior points of a subset of $X$.