Examples and counterexamples[edit]

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 form a countable local base at $x.$

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

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 $\left(x_{n}\right)_{n=1}^{\infty }$ in $A$ that converges to $x.$ (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 $f$ 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 $x_{n}\to x,$ where $x_{n}\neq x$ for all $n,$ we have $f\left(x_{n}\right)\to L.$ Also, if $f$ is a function on a first-countable space, then $f$ is continuous if and only if whenever $x_{n}\to x,$ then $f\left(x_{n}\right)\to f(x).$

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