Page 161 - DMTH503_TOPOLOGY

Unit 17: The Separation Axioms

Finally {x}, {y} are closed sets in X. Notes
Generalising this result.
{x} is closed  x  X.

Example 3: Prove that in a T -space all finite sets are closed.
1
Solution: Let (X, T) be a T -space.
1
To prove that {x} is closed  x  X.
Now write (ii) part of the proof of the theorem 1
Let A be an arbitrary finite subset of X.

Then A =  {{x}} : x  A}
= finite union of closed sets = closed set.
 A is a closed set.

Example 4: A topological space (X, T) is a T -space iff T contains the cofinite topology
1
on X.
Solution: Let (X, T) be a T -space.
1
To prove that T contains cofinite topology on X, we have to show that T contains subsets A of X
s.t. X – A is finite.

Here we shall make use of the fact that
X is T -space  {x} is closed  x  X
1
 X – {x} is open subset of X  X – {x}  T

Thus X – {x}  T  X – (X – {x}) = {x} = finite set.
This is true  x  X.
Hence by definition T contains cofinite topology on X.

Conversely, suppose that T contains cofinite topology on X.
To prove that (X, T) is T -space.
1
{x} is a finite subset of X.
Also T contains cofinite topology.

Consequently X – {x}  T so that
{x} is closed  x  X
 (X, T) is T -space.
1
Theorem 2: A topological space X is a T -space of X iff every singleton subset {x} of X is closed.
1
Proof: Let X be a T -space and x  X.
1
By the T -axiom, we know that if y  x  X, than there exists an open set G which contain y but
1 y
not x i.e.
y = G  {x} c
y

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