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 X there exists a sequence U₁, U₂, … of open neighbourhoods of x such that for any open neighbourhood 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 first 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. Since ω₁ is the only such point, however, the subspace ω₁ = [0,ω₁) is first-countable.

The quotient space $\mathbb {R} /\mathbb {N}$ 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.

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.

See also

• Second-countable space
• Separable space

References

• "first axiom of countability", Encyclopedia of Mathematics, EMS Press, 2001 [1994]