Page 167 - DMTH503_TOPOLOGY

Unit 17: The Separation Axioms

Proof: Let (X, T) be a first countable space in which every convergent sequence has a unique Notes
limit. If possible, let (X, T) be not a Hausdorff space.

Then given x, y  X, x  y,  open sets G and H
such that x  G, y  H, G H  
Now (X, T) being first countable, there exists monotone decreasing local bases
 = {B (x) : x  N} and
x n
 = {B (y) : n  N} at x and y respectively.
y n
Clearly, B (x) B (y)    n  N
n n
[ B (x) and B (y) are open nhds. of x and y respectively]
n n
Let x  B (x) B (y)  n  N
n n n
But B (x) and B (y) being monotone decreasing local bases at x and y respectively,  a positive
n n
integer n such that
0
n > n  B (x)  G and
0 n
B (y)  H
n
 x  B (x)  G and
n n
x  B (y)  H
n n
 x  G and x  H
n n
 x  x and x  y
n n
But, this contradicts the fact that every convergent sequence in X has a unique limit.
Hence, (X, T) must be a Hausdorff space.

Theorem 9: The product space of two Hausdorff spaces is Hausdorff.
Proof: Let X and Y be two Hausdorff spaces. We shall prove that X × Y is also a Hausdorff spaces.
Let (x , y ) and (x , y ) be any two distinct points of X × Y.
1 1 2 2
Then either x  x or y  y
1 2 1 2
Let us take x  x
1 2
Since X is a Hausdorff space,  T open nhds. G and H of x and x respectively such that x  G, x
1 2 1 2
 H and G H = 
Then G × Y and H × Y are open subsets of X × Y such that
(x , y )  G × Y,
1 1
(x , y )  H × Y and
2 2
(G × Y) (H × Y) = (G H) × Y
=  × Y = 

Thus, in this case, distinct points (x , y ) and (x , y ) of X × Y have disjoint open nhds.
1 1 2 2
Similarly, when y  y  disjoint open nhds of (x , y ) and (x , y )
1 2 1 1 2 2
Hence X × Y is Hausdorff.

LOVELY PROFESSIONAL UNIVERSITY 161