# 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 ${\displaystyle X}$ is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point ${\displaystyle x}$ in ${\displaystyle X}$ there exists a sequence ${\displaystyle N_{1},N_{2},\ldots }$ of neighbourhoods of ${\displaystyle x}$ such that for any neighbourhood ${\displaystyle N}$ of ${\displaystyle x}$ there exists an integer ${\displaystyle i}$ with ${\displaystyle N_{i}}$ contained in ${\displaystyle N.}$ 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 ${\displaystyle x}$  with radius ${\displaystyle 1/n}$  for integers form a countable local base at ${\displaystyle x.}$

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 ${\displaystyle \omega _{1}+1=\left[0,\omega _{1}\right]}$  where ${\displaystyle \omega _{1}}$  is the first uncountable ordinal number. The element ${\displaystyle \omega _{1}}$  is a limit point of the subset ${\displaystyle \left[0,\omega _{1}\right)}$  even though no sequence of elements in ${\displaystyle \left[0,\omega _{1}\right)}$  has the element ${\displaystyle \omega _{1}}$  as its limit. In particular, the point ${\displaystyle \omega _{1}}$  in the space ${\displaystyle \omega _{1}+1=\left[0,\omega _{1}\right]}$  does not have a countable local base. Since ${\displaystyle \omega _{1}}$  is the only such point, however, the subspace ${\displaystyle \omega _{1}=\left[0,\omega _{1}\right)}$  is first-countable.

The quotient space ${\displaystyle \mathbb {R} /\mathbb {N} }$  where the natural numbers on the real line are identified as a single point is not first countable.[1] However, this space has the property that for any subset ${\displaystyle A}$  and every element ${\displaystyle x}$  in the closure of ${\displaystyle A,}$  there is a sequence in A converging to ${\displaystyle x.}$  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 ${\displaystyle A,}$  a point ${\displaystyle x}$  lies in the closure of ${\displaystyle A}$  if and only if there exists a sequence ${\displaystyle \left(x_{n}\right)_{n=1}^{\infty }}$  in ${\displaystyle A}$  that converges to ${\displaystyle x.}$  (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 particular, if ${\displaystyle f}$  is a function on a first-countable space, then ${\displaystyle f}$  has a limit ${\displaystyle L}$  at the point ${\displaystyle x}$  if and only if for every sequence ${\displaystyle x_{n}\to x,}$  where ${\displaystyle x_{n}\neq x}$  for all ${\displaystyle n,}$  we have ${\displaystyle f\left(x_{n}\right)\to L.}$  Also, if ${\displaystyle f}$  is a function on a first-countable space, then ${\displaystyle f}$  is continuous if and only if whenever ${\displaystyle x_{n}\to x,}$  then ${\displaystyle f\left(x_{n}\right)\to f(x).}$

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 ${\displaystyle \left[0,\omega _{1}\right).}$  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.