Page 13 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

Notes

Note In every topological space, X and  are open as well as closed.

1.3.2 Door Space

A topological space (X, T) is said to be a door space if every subset of X is either T-open or
T-closed.

Example 10: Let X = {1, 2, 3) and T = {, {1, 2}, {2, 3}, {2}, X}
Then, T-closed sets are X, {3}, {1}, {1, 3}, .
This shows that every subset of X is either T-open or T-closed.

1.3.3 Closure of a Set

Let (X, T) be a topological space and A is a subset of X, then the closure of A is denoted by A or
Cl (A) is the intersection of all closed sets containing A or all closed superset of A.

Example 11: If T = {, {a}, {a, b}, {a, c, d}, {a, b, c}, {a, b, c, d}, X} be a topology on X = {a, b,
c, d, e} then find the closure of the sets {a}, {b}
Solution: Closed subset of X are
, {a}, {a, b}, {a, c, d}, (a, b, e}, {a, b, c, d}, X = X, {b, c, d, e}, {c, d, e}, {b, e}, {c, d}, {e}, 
then { a } = X

{ b } = X  {b, c, d, e}  {b, e} = {b, e}
Theorem 3: A is closed iff A = A
Proof: Let us suppose that A is closed

 A  A (by definition of closure)
Now also A  A (A is common in all supersets of A)
 A = A
Conversely, let us suppose that A = A
Since we know that A is closed. (by definition of closure of A)

 A = A is closed
 A is closed

1.3.4 Properties of Closure of Sets

Theorem 4: Let (X, T) be a topological space and let A, B be any two subsets of X. Then

(i)  = , X = X;
(ii) A  A

(iii) A  B  A  B

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