# 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}\$\$

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