Page 164 - DMTH503_TOPOLOGY
P. 164
Topology
Notes Thus, G and H are open sets such that
x G but y G
and y H but x H
Hence, the space (X, T) is a T -space.
1
Conversely, let us consider the cofinite topology T on an infinite set X.
Let x be an arbitrary point of X.
by definition of T,
X – {x} is open, for {x} is finite set and so {x} is T-closed.
Thus, every singleton subset of X is closed.
It follows that the space (X, T) is a T -space. Now we shall show that the space (X, T) is not a
1
T -space.
2
For this topology, no two open subsets of X can be disjoint.
Let if possible G and H be two open disjoint subsets of X, then
G H =
(G H)’ = ’
G’ H’ = X (by De-Morgan’s law)
Here G’ H’ being the union of two finite sets is finite, where as X is infinite.
Hence for this topology no two open sets can be disjoint i.e. no two distinct points can be
separated by open sets.
Hence, (X, T) is not T -space.
2
Theorem 5: Every subspace of a T -space is a T -space
2 2
or
Prove that every subspace of a Hausdorff space is also Hausdorff.
Proof: Let (X, T) be a Hausdorff space and (Y, T ) be a subspace of it.
y
Let x and y be any two distinct points of Y.
Then x and y are distinct points of X.
But (X, T) is a Hausdorff space, T-open nhds. G and H of x and y respectively such that
G H =
Consequently, Y G and Y H are T -open nhds of x and y respectively.
y
Also x G, x Y x Y G
and y H, y y y Y H
and since G H = , we have
(Y G) (Y H) = Y (G H) = Y =
This shows that (Y, T ) is also a T -space. Hence, every subspace of a Hausdorff space is also a
y 2
Hausdorff space.
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