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 exist an open set $U$ such that

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 union of all open subsets of $A$, and a point in the interior of $A$ is called an interior point of $A$.

Remarks:
• Interior of $A$ is union of all open sets contained in $A$. Union of open sets is again an open set. Hence interior of $A$ is the largest open set contained in $A$.
${\phi ^o} = \phi$ and ${X^o} = X$
• 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 n 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 subset 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) ${\left( {A \cup B} \right)^o} \supseteq {A^o} \cap {B^o}$