Page 146 - DMTH503_TOPOLOGY

Topology

Notes
 (Aº)  A  A
 (Aº)  A ...(1)

 x  Aº  A  G

 x  Aº  (Aº)  A  G, [by (1)]

Taking Aº = U

x  U  U  G
Thus we have shown that given any nbd G of x,  a nbd U of x s.t.

x  U  U  G
Consequently X is regular.

Theorem 3: Any open subspace of a locally compact space is a locally compact.
Proof: Let (Y, U) be an open subspace of a locally compact space (X, T) so that Y is open in X.

To prove that Y is locally compact.
Let x  Y  X be arbitrary and G a U-nbd of x in Y, then x  X, G  Y.
X is locally compact  X is locally compact at x.

G is a U-nbd of x in Y   G  U s.t. x  G  G
1 1
 G  T s.t. x  G  G. For Y is open in X.
1 1
 G is a T-nbd of x in X.
Also X is locally compact   a compact set A  X s.t. x  Aº  A  G. But G  Y.
 x  Aº  A  G  Y

Thus (i) A  Y, A is U-compact.
For A is T-compact  A is U-compact.

(ii) G is a nbd of x in Y s.t. x  Aº  A  G.
This proves that Y is locally compact at any y  Y and hence the result follows. Proved.
Theorem 4: Every closed subspace of a locally compact space is locally compact.

Proof: Let (Y, U) be a closed subspace of a locally compact space (X, T), then Y is T-closed set. Let
y  Y  X be arbitrary.
To prove that Y is locally compact, we have to prove that Y is locally compact at y.

X is locally compact  X is locally compact at y

  T-open nbd N of x s.t. N is T-compact.
 N Y is U-open nbd of y.

N Y  N  N Y  N.

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