# 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 *N*_{1}, *N*_{2}, … of neighbourhoods of *x* such that for any neighbourhood *N* of *x* there exists an integer *i* with *N*_{i} contained in *N*.
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 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 ω_{1}+1 = [0,ω_{1}] where ω_{1} is the first uncountable ordinal number. The element ω_{1} is a limit point of the subset [0,ω_{1}) even though no sequence of elements in [0,ω_{1}) has the element ω_{1} as its limit. In particular, the point ω_{1} in the space ω_{1}+1 = [0,ω_{1}] does not have a countable local base. Since ω_{1} is the only such point, however, the subspace ω_{1} = [0,ω_{1}) is first-countable.

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.

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

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,ω_{1}). 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

- Hazewinkel, Michiel, ed. (2001), "first axiom of countability",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4