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) $${left( {A cup B} right)^o} supseteq {A^o} cap {B^o}$$

Source Article