Page 25 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

13. Let X = {a, b, c} and let T = {, X, {a}, {b}, {a, b}}, show that D({a}) = {c}, D({c}) =  and find Notes
derived sets of other subsets of X.

1.7 Interior and Exterior

1.7.1 Interior Point and Exterior Point

Interior Point: Let X be a topological space and let A  X.

A point x  A is called an interior point of A iff  an open set G such that x  G  A.
The set of all interior points of A is known as the interior of A and is denoted by Int (A) or A°.
Symbolically,

A° = Int (A) =  {G  T : G  A}.

Example 26: Let T = {, {a}, {b, c}, {a, b}, {a, b, c}, X} be a topology on X = {a, b, c, d} then
Int (A) = Union of all open subsets of X which are contained in A.
Int [{a}] =   {a} = {a}
Int [{a, b}] =   {a}  {a, b} = {a, b}
Exterior Point: Let X be a topological space and let A  X.

C
A point x  A is called an exterior point of A iff it is an interior point of A or X – A.
The set of all exterior points of A is called the exterior of A and is denoted by ext (A).
Symbolically,
ext (A) = (X – A)° or (A )°.
C

Example 27: Let T = {X, , {p}, {p, q}, {p, q, t}, {p, q, r, s}, {p, r, s}} be the topology on X =
{p, q, r, s, t}
Determine exterior of (i)
B = {q}

Solution: ext (B) = (X – B)° = {p, r, s, t}°
=  {Q, {p}, {p, r, s}}
= {p, r, s}

1.7.2 Interior Operator and Exterior Operator

Interior Operator: Let X be a non-empty set and P(X) be its power set. Then, an interior operator
‘i’ on X is a mapping i : P(X)  P(X) which satisfies the following four axioms:
(i) i(X) = X
(ii) i(A)  A
(iii) i(A  B) = i(A)  i(B)

(iv) i(i(A) = i(A), where A and B are subsets of X.

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