Page 166 - DMTH503_TOPOLOGY

Topology

Notes
Example 8: Show that every convergent sequence in Hausdorff space has a unique limit.
Solution: Let (X, T) be a Hausdorff space.

Let <x > be a sequence of points of Hausdorff space X.
n
Let Lt x = x
n n
Suppose, if possible,

Lt x = y, where x  y.
n n
Since X is a Hausdorff space,  open sets G and H such that x  G, y  H
and G H =  ...(1)
Since x  x and x  y
n n
and G, H are nhds of x and y respectively,  positive integers n and n such that
1 2
x  G  n  n and
n 1
x  H  n  n
n 2
Let n = max (n , n ), then x  G H  n  n
0 1 2 n 0
This contradicts (1).
Hence, the limit of the sequence must be unique.

Note Converse of the above theorem is not true.

Example 9: Show that each singleton subset of a Hausdorff space is closed.

Solution: Let X be a Hausdorff space and let x  X.
Let y  X be any arbitrary point of X other than x i.e. x  y.
Since X is a T -space,  a nhd of y which does not contain x.
2
It follows that y is not a limit point of {x} and consequently D({x}) = 

Hence   x = x.

This shows that {x} is T-closed.

Example 10: Show that every finite T -space is discrete.
2
Solution: Let (X, T) be a finite T -space. We know that every singleton subset of X is T-closed. Also
2
a finite union of closed sets is closed. It follows that every finite subset of X is closed.
Hence, the space is discrete.
Theorem 8: A first countable space in which every convergent sequence has a unique limit is a
Hausdorff space.

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