Page 165 - DMTH503_TOPOLOGY

Unit 17: The Separation Axioms

Theorem 6: The property of being a Hausdorff space is a topological invariant. Notes
or
The property of being a Hausdorff space is preserved by one-one onto open mapping and hence
is a topological property.
Proof: Let (X, T) be a T -space and let (Y, T ) be any topological space.
2 y
Let f be a one-one open mapping of X onto Y. Let y , y be two distinct elements of Y. Since f is
1 2
one-one onto map, there exists distinct elements x and x of X such that y = f(x ) and y = f(x ).
1 2 1 1 2 2
Since (X, T) is a T -space,  T-open nhds. G and H of x and x such that G H = 
2 1 2
Now, f being open, it follows that f(G) and f(H) are open subsets of Y such that
y = f(x )  f(G)
1 1
y = f(x )  f(H)
2 2
and f(G) f(H) = f(G H) = f() = 
This shows that (Y, T ) is also a T -space.
y 2
Since a property being a T -space is preserved under one-one, onto, open maps, it is preserved
2
under homeomorphism.
Hence, it is a topological property.
Theorem 7: Prove that every compact subset of Hausdorff space is closed.

Proof: Let (Y, T*) be a compact subset of Hausdorff space (X, T).
In order to prove that Y is T-closed, we have to show that X – Y is T-open.
Let x be an arbitrary element of X – Y.
Since (X, T) is a T -space, then for each y  Y,  T-open sets G and H such that
2 y y
x  G , y  H and G H = 
y y y y
Now consider the class
 = {H Y : y  Y}
y
Clearly,  is T*-open cover of Y.

Since (Y, T*) is a compact subset of (X, T), there must exist a finite sub cover of  i.e.  n points
y , y , ..., y in Y such that
1 2 n
{H Y : i  T } is a finite sub cover of .
yi n
n
Thus Y  {H }
i y
i 1

n  n 
Let N = {G }, then N is T-nhd of x, and N  {H }   .
i y 
yi


i 1  i 1 
Thus, N Y =   N  X – Y
i.e. X – Y contains a T-nhd of each of its points.
Hence, X – Y is T-open i.e. Y is T-closed.

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