Page 21 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

Notes
Example 19: Let X = {1, 2, 3, 4, 5} be a non-empty set and T = {, X, {3}, {3, 4}, {2, 3}, {2, 3, 4}}
is a topology defined on X. Suppose a subset A = {1, 3, 5}  X. The closed set are:
X, , {1, 2, 4, 5}, {1, 2, 5}, {1, 4, 5}, {1, 5}.

So, we have A = X. Since A is finite and dense in X. So X is a separable space.
Theorem 10: Show that the cofinite topological space (X, T) is separable.
Solution: Let (X, T) be a cofinite topological space.

(i) When X is countable.
Then X  X and X = X
This shows that X is separable.

(ii) Let A  X s.t. A is finite.
By definition of cofinite topological space A = X – A is open so that A is closed.
 every finite set A is T-closed and so A = X.

Now A = X, A is countable.
This shows that (X, T) is separable.

Example 20: A discrete space X is separable iff X is countable.

Solution: As we know that every subset of a discrete space (X, T) is both open and closed. Also, A
is said to be everywhere dense in X if A = X.
Also, X is separable if  A  X s.t. A = X and A is countable.

So, the only everywhere dense subset of X is X itself.
 X can have a countable dense subset iff X is countable.
Hence, X is separable iff X is countable.

1.6.2 Limit Point or Accumulation Point or Cluster Point

Let (X, T) be a topological space and A  X. A point x  X is said to be the limit point or accumulation
point or cluster point of A if each open set containing ‘x’ contains at least one point of A different
from x.
Thus, it is clear from the above definition that the limit point of a set A may or may not be the
point of A.

Example 21: Let X = {a, b, c} with topology

T = {, {a, b}, {c}, X} and A = {a}, then b is the only limit point of A, because the open sets containing
b namely {a, b} and X also contains a point of A.
Whereas, ‘a’ and ‘b’ are not limit point of C = {c}, because the open set {a, b} containing these
points do not contain any point of C. The point c is also not a limit point of C, since then open set
{c} containing ‘c’ does not contain any other point of C different from c. Thus, the set C = {c} has
no limit points.

LOVELY PROFESSIONAL UNIVERSITY 15