Page 152 - DMTH503_TOPOLOGY

Topology

Notes Let T be a discrete topology on an infinite set X so that every subset of X is open in X and hence
in, particular, each singleton set {x} is open in X for each x X.

Write  = {{x} : x X}.
Then it is easy to verify that  is a base for the topology T on X and  is not countable. For X is
not countable. Hence X is not second countable.

If we take L = {x} then evidently L is a countable local base at x  X as it has only one number.
x x
For any G  T with x  G, {x} s.t. x  {x}  G. From what has been done, it follows that X is first
countable but not second countable.

Theorem 5: Show that the property of a space being first countable is hereditary.
Proof: Let (Y, ) be a subspace of a first countable space (X, T).
If we show that (Y, ) is first countable, we can conclude the required result.

Let y Y be arbitrary, then y X. [ Y  X]
X is first countable  a countable local base at each x X and hence, in particular, a countable
local base  at y X.
Members of  can be enumerated as B , B , B , B , ...
1 2 3 4
i.e.  = {B : n N}.
n n
Evidently, y  B  n N.
n
Write B = {Y B : n N} ...(1)
1 n
y Y, y B  n N  y  Y B  n N ...(2)
n n
B    n N  B T Y B  ...(3)
n n n
We claim  is a countable local base at y for on Y.
1
(i) Evidently N ~B under the map n Y B . Hence B is countable. ...(4)
1 n 1
(ii) any G  y G ...(5)
1
(iii)  is family of all  - open sets. ...(6)
1
(iv) let G  be arbitrary s.t.

y G, then H T s.t. G = H Y.
y H. For y G = H Y.
By definition of local base.
y H T   B   s.t. y B H
r r
or y  H Y   B Y   s.t. B Y H Y
r 1 r
or y  G   B  Y   s.t. y  B Y G ...(7)
r 1 r
The result (1), (4), (5), (6) and (7) taken together imply that  is a local base at y Y for the
1
topology on Y and hence (Y, ) is first countable.
Theorem 6: Show that the property of a space being second countable is hereditary.
or
Prove that every subspace of a second countable space is second countable.

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