Page 162 - DMTH503_TOPOLOGY

Topology

Notes Then {x} =  {y : y  x}  {G : y  x}  {x} .
c
c
y
c
Therefore {x} =  {G : y  x}.
y
c
Thus {x} being the union of open sets is an open set. Hence {x} is a closed set.
Conversely, let us suppose that {x} is closed.
We have to prove that X is a T -space.
1
Let x and y be two distinct points of X.
Since {x} is a closed set, {x} is an open set which contains y but not x.
c
c
Similarly {y} is an open set which contains x but not y.
Hence X is a T -space.
1
Theorem 3: The property of being a T -space is preserved by one-to-one onto, open mappings
1
and hence is a topological property.
Proof: Let (X, T) be a T -space and let (Y, V) be a space homomorphic to the topological space
1
(X, T).
Let f be a one-one open mapping of (X, T) onto (Y, V).

We shall prove that (Y, V) is also a T -space.
1
Let y , y be any two distinct points of y.
1 2
Since the mapping f is one-one onto, there exist, points x and x in X such that
1 2
x  x and f(x ) = y and f(x ) = y
1 2 1 1 2 2
Since (X, T) is a T -space, there exists T-open sets G and H such that
1
x  G but x  G
1 2
x  H but x  H
2 1
Again, since f is an open mapping, f[G] and f[H] are V-open subsets such that

f(x)  f[G] but f(x )  f[G]
2
and f(x )  f[H] but f(x )  f[x]
2 1
Hence (Y, V) is also a T -space.
1
Thus, the property of being a T -space is preserved under one-one onto, open mappings.
1
Hence it is a topological property.
Theorem 4: Every subspace of T -space is a T -space i.e. the property being a T -space is hereditary.
1 1 1
Proof: Let (X, T) be a T -space and let (X*, T*) be a subspace of (X, T).
1
Let x and x be two distinct point of X*. Since X*  X, x and x are also distinct points of X. But
1 2 1 2
(X, T) is a T -space, therefore there exist T-open sets G and H such that
1
x  G but x  G
1 2
and x  H but x  H
2 1
Then G = G X*
1
and H = H X* are T*-open sets such that
1
X  G but x  G
1 1 2 1

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