※ 引述《jasonni ()》之銘言：
: 想請教幾題有關拓樸空間基本的証明
: 1.prove that every countable first-countable space is second countable
Let X be a countable, first-countable space.
For all x belongs to X, there exists a sequence of local basis Ux1,Ux2,...
s.t. for any open set V containing x, there exists k satisfying x belongs to
Uxk and Uxk is contained in V.
Let S be an non-empty open set in X.
Since X is countable, S can be written as {s1,s2,....}
For each si, there exists a sequence of local basis U(si)1,U(si)2,...
In particular, for the set S we can choose U(si)(ki) s.t. si belongs to
U(si)(ki) and U(si)(ki) is contained in S.
Taking union, we see that the union of {U(s1)(k1),U(s2)(k2),....} is contained
in S, since each of them is contained in S.
Also, S is contained in the union of {U(s1)(k1),U(s2)(k2),....}, since each
element of S belongs to some of the U(si)(ki).
Thus S=the union of {U(s1)(k1),U(s2)(k2),....}
We conclude that any open set in X can be written as a union of those local
basis.
The total cardinality of those local basis is countable, since countable union
of countable sets is countable.
Hence X is second-countable.
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