First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space, "X", is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point, "x", in space "X" there exists a sequence, "U"₁, "U"₂, … of open neighborhoods of "x" such that for any open neighborhood, "V", of "x", there exists an integer, "i", with "U"_"i" contained in "V".

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 = [0,ω₁] where ω₁ is the smallest uncountable ordinal number. The element ω₁ is a limit point of the subset [0,ω₁) even though no sequence of elements in [0,ω₁) has the element ω₁ as its limit. In particular, the point ω₁ in the space ω₁+1 = [0,ω₁] does not have a countable local base. The subspace ω₁ = [0,ω₁) is first-countable however, since ω₁ is the only such point.

Properties

One of the most important properties of first-countable spaces is that given a subset "A", a point "x" lies in the closure of "A" if and only if there exists a sequence {"x"_"n"} in "A" which converges to "x". This has consequences for limits and continuity. In particular, if "f" is a function on a first-countable space, then "f" has a limit "L" at the point "x" if and only if for every sequence "x"_"n" → "x", where "x"_"n" ≠ "x" for all "n", we have "f"("x"_"n") → "L". Also, if "f" is a function on a first-countable space, then "f" is continuous if and only if whenever "x"_"n" → "x", then "f"("x"_"n") → "f"("x").

In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω₁). Every first-countable space is compactly generated.

Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be.

ee also

*Second-countable space
*Separable space