Frontier of a set A, where A is a Subset of a topological space (X, T) is denoted by Fr(A) and is defined as

Fr(A)=cl(A)∩cl(A')

Fr(A)=cl(A)∩cl(X-A)

Frontier of a Set Topology |

Fr(A’)=cl(A')∩cl((A')')

Fr(X-A)=cl(X-A)∩cl(X-(X-A))

Fr(X-A)=cl(X-A)∩cl(A)

In others words frontier of a set contains all the points which are in the closure of a set but not in its interior.

By definition frontier of the complement of a set is same as the frontier of that set.

## Boundary of a Set Topology

Boundary of a set A is denoted by ∂(A) and is defined as the set of all boundary points of a set A.

In others words the set of all points which are neither entirely from the set and nor entirely outside the set. Boundary of a set consists of its outer most limits points.

## Boundary Point of a Set

A point x is called boundary point of a set if every neighborhood of x contains both points from the set and the points outside the set.

### Boundary Or Frontier of a Set in Topology Examples:

#### 1) Let X={1,2,3,4} and T={Φ,X,{1}} be a topology defined on X. If A={1},where A⊆X then find Fr(A) or ∂(A).

**Solution;**

• cl(A)=?

All closed sets are

Φ,X and {1}

All closed super sets of A are

X, {1}

cl(A)={1}∩X={1}

• cl(A’)=cl(X-A)=?

X-A={2,3,4}

All closed sets are

Φ,X and {1}

All closed super sets of (X-A)=(A’) is

X

cl(X-A) = X

• Now we know that

Fr(A)=∂(A)=cl(A)∩cl(A’)

Fr(A)=∂(A)={1}∩X={1}

#### 2) Let X={1,3,5,7} and T={Φ,X,{1,5}} be a topology defined on X. If A={5},where A⊆X then find Fr(A) or ∂(A).

**Solution;**

• cl(A)=?

All closed sets are

Φ,X and {1,5}

All closed super sets of A are

X, {1,5}

cl(A)={1,5}∩X={1,5}

• cl(A’)=cl(X-A)=?

X-A={1,3,7}

All closed sets are

Φ,X and {1,5}

All closed super sets of (X-A)=(A’) is

X

cl(X-A) = X

• Now we know that

Fr(A)=∂(A)=cl(A)∩cl(A’)

Fr(A)=∂(A)={1,5}∩X={1,5}

## Frequently Asked Questions:

### What is the difference between boundary and frontier of a set?

There is no difference between the boundary and frontier of a because both boundary and frontier of a set consists of its all the outermost limits points.