# Interior Point of a Set

Let $$\left( {X,\tau } \right)$$ be the topological space and $$A \subseteq X$$, then a point $$x \in A$$ is said to be an interior point of set $$A$$, if there exists an open set $$U$$ such that

\[x \in U \subseteq A\]

In other words let $$A$$ be a subset of a topological space $$X$$, a point $$x \in A$$ is said to be an interior points of $$A$$ if $$x$$ is in some open set contained in $$A$$.

**Interior of a Set**

Let $$\left( {X,\tau } \right)$$ be a topological space and $$A$$ be a subset of $$X$$, then the interior of $$A$$ is denoted by $${\text{Int}}\left( A \right)$$ or $${A^o}$$ is defined to be the union of all open sets contained in $$A$$.

In other words let $$\left( {X,\tau } \right)$$ be a topological space and $$A$$ be a subset of $$X$$. The interior of $$A$$ is the union of all open subsets of $$A$$, and a point in the interior of $$A$$ is called an interior point of $$A$$.

**Remarks:**

• The interior of $$A$$ is the union of all open sets contained in $$A$$. The union of open sets is again an open set. Hence the interior of $$A$$ is the largest open set contained in $$A$$.

• $${\phi ^o} = \phi $$ and $${X^o} = X$$

• The interior of sets is always open.

• $${A^o} \subseteq A$$

**Example:**

Let $$X = \left\{ {a,b,c,d,e} \right\}$$ with topology $$\tau = \left\{ {\phi ,\left\{ b \right\},\left\{ {a,d} \right\},\left\{ {a,b,d} \right\},\left\{ {a,c,d,e} \right\},X} \right\}$$. If $$A = \left\{ {a,b,c} \right\}$$, then find $${A^o}$$. Since there is no open set containing $$a$$ and a subset of $$A$$, so $$a$$ is not an interior point of $$A$$. Similarly, $$c$$ is not an interior point of $$A$$. Since $$\left\{ b \right\}$$ is an open set containing $$b$$ and is a subset of $$A$$, so $$b$$ is an interior point of $$A$$. Hence $${A^o} = \left\{ b \right\}$$.

**Theorems**

• Each point of a non empty subset of a discrete topological space is its interior point.

• The interior of a subset of a discrete topological space is the set itself.

• The interior of a subset $$A$$ of a topological space $$X$$ is the union of all open subsets of $$A$$.

• The subset $$A$$ of topological space $$X$$ is open if and only if $$A = {A^o}$$.

• If $$A$$ is a subset of a topological space $$X$$, then $${\left( {{A^o}} \right)^o} = {A^o}$$.

• Let $$\left( {X,\tau } \right)$$ be a topological space and $$A$$ and $$B$$ are subsets of $$X$$, then **(1)** $$A \subseteq B \Rightarrow {A^o} \subseteq {B^o}$$ **(2)** $${\left( {A \cap B} \right)^o} = {A^o} \cap {B^o}$$ **(3)** $${\left( {A \cup B} \right)^o} \supseteq {A^o} \cap {B^o}$$