In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability is said to be first-countable if each point has a countable neighbourhood basis there exists a sequence of neighbourhoods contained in Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.

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.

An example of a space that is not first-countable is the cofinite topology on an uncountable set (such as the real line). More generally, the Zariski topology on an algebraic variety over an uncountable field is not first-countable.

Another counterexample is the ordinal space is the first uncountable ordinal is a limit point is first-countable.

The quotient space where the natural numbers on the real line are identified as a single point is not first countable. A space with this sequence property is sometimes called a Fréchet–Urysohn space.

First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, but any uncountable discrete space is first-countable but not second-countable.

Properties

lies in the closure if and only if there exists a sequence that converges (In other words, every first-countable space is a Fréchet-Urysohn space and thus also a sequential space.) This has consequences for limits and continuity

In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces that are not compact (these are necessarily not metrizable spaces). One such space is the ordinal space 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

References

Bibliography