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)
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| 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.