Page 20 - DMTH503_TOPOLOGY

Topology

Notes
Example 17: Prove that every non-empty subset of an indiscrete space is dense in X.
Solution: Let (X, T) be an indiscrete space.

Let A  X be non-empty set.
To show: A is dense in X.
For this, we are to prove A = X
By definition of an indiscrete topology,
T = {, X}
T-open sets are , X
T-closed sets are X – , X i.e. X, .
Since A   by assumption.
 The only closed superset of A is X,

so that A = X.

Example 18: 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 boundary of the following sets
(i) B = {q}
B ° =  {} = 
(X – B)° = {p, r, s, t}° =  {f, {p}, {p, r, s}}

= {p, r, s}
b(B) = X – B°  (X – B)°
= X –   {p, r, s}
= {q, t}

Self Assessment

10. In a topological space, prove that:
(i) A is dense  it intersects every non-empty open set.

(ii) A is closed  A contains its boundary.
11. In any topological space, prove that
b(A) =   A is open as well as closed.

1.6 Separable Space, Limit Point and Derived Set

1.6.1 Separable Space

Let X be a topological space and A be subset of X, then X is said to be separable if

(i) A = X
(ii) A is countable

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