Examples and Counterexamples
The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers n > 0 form a countable local base at x.
An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).
Another counterexample is the ordinal space ω1+1 = where ω1 is the first uncountable ordinal number. The element ω1 is a limit point of the subset does not have a countable local base. The subspace ω1 = [0,ω1) is first-countable however, since ω1 is the only such point.
The quotient space where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a Fréchet-Urysohn space.
Read more about this topic: First-countable Space
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