Interior of a set A in topological space (X, T), where A is the subset of a non empty set X with topology T defined on it, is denoted by A° and is defined as “the union of all open sets which are contained in A".
In other words, the largest open set which is contained in A.
 |
| Interior of a set in topololgy |
Examples:
1) Let X = {1,3,5,7} be a non empty set and T = {Φ,X,{1},{3,5},{7},{1,3,5},{1,7},{3,5,7}} is a topology defined on it. If A = {1,5,7} is a subset of X then find interior of A.
Solution;
• All open sets are
Φ,X,{1},{3,5},{7},{1,3,5},{1,7} and {3,5,7}
• All open sets which are contained in A are
Φ,{1},{7} and {1,7}
• Union of all open sets which are contained in A is
A° = Φ∪{1}∪{7}∪{1,7} = {1,7}
⇒ A° = {1,7} is the largest open set which is contained in A.
2) Let X = {1,3,5} be a non empty set and T = {Φ,X,{1},{3,5},{5},{1,5}} is a topology defined on it. If A = {1,5} is a subset of X then find interior of A.
Solution;
• All open sets are
Φ,X,{1},{3,5},{5},{1,5}
• All open sets which are contained in A are
Φ,{1},{5} and {1,5}
• Union of all open sets which are contained in A
A° = Φ∪{1}∪{5}∪{1,5} = {1,5}
Frequently Asked Questions:
How to find interior of a set in topology?
To find interior of A ,the following algorithm is used;
• Firstly, find All open Sets.
• Secondly, find all those open sets which are contained in A.
• At last, take union of all open sets which are contained in A.
• Union of all open sets which are contained in A is the largest open set which is contained in A and is called interior of A and is denoted by A°.
What is the interior of a set in discrete topology?
Interior of a discrete topology is the parent set on which topology is defined because every set of discrete topology is both open and closed and the largest open set present in discrete topology is parent set.
What is the interior of a set in indiscrete topology?
Interior of a set in indiscrete topology is parent set on which indiscrete topology is defined because the members of indiscrete topology are only empty set and parent set which are both open and closed.