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10.3 Definition. A topological space X is first countable (or satisfies the first axiom of countability) if each x e X has a countable nhood base. Since the disks about x of rational radius form a...

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10.3 Definition. A topological space X is first countable (or satisfies the first axiom of countability) if each x e X has a countable nhood base. Since the disks about x of rational radius form a nhood base at x in any pseudometric space, the pseudometrizable spaces are all first countable. They form the most important single class of first-countable spaces. The first axiom of countability has been defined before, in 4.4(b), but you may have missed it. The second axiom was introduced in 5F. Both will be studied in detail in Section 16.
10.4 Theorem. If X is a first countable space and E c X, then x e E if there is a sequence (x„) contained in E which converges to x.

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Answered Same Day Dec 20, 2021

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Robert answered on Dec 20 2021
114 Votes
X: uncountable set with Cofinite topology
1. If (xn) is a sequence, we claim that (xn) converges if and only if there is
atmost one term of sequence which occurs infinitely times. Proof is the
following:
Suppose (xn) is a convergent sequence and there are a and b such that
xn = a, b for infinite values of n. Then take U = X − b open set around
a, it does not contain infinitely point of xn, because we assumed cn =
for infinite times.
So this proves that xn can not converge to a. Same argument shows that
xn can not converge to b. So suppose that there is c such that xn → c. But
then U −{a, b} is open neighborhood of c, which does not contain infinite
point of xn. Hence xn can not converge, But this is a contradiction. So if
(xn) converges then it can take atmost one value infinite times.
Now conversely, suppose (xn) take only atmost one value infinite times
then we claim (xn) converges.
Suppose xn takes b infinite times then if V is an open neighborhood of
then X − V is finite. This implies X − V may contains only finite terms
of (xn). Hence we see that xn ∈ V ∀n but finite. That is xn → b.
On the other hand if no values is taken by (xn) infinite times then (xn)
will converge to every point of X.
Note: When we are saying that xn takes infinite times b this means xn =
for infinite values of n. When we say that no values is...
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