First-countable space

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 which is not first-countable is the cofinite topology on an uncountable set (such as the real line).

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.

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 ${\displaystyle A}$ which 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 which 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.

• Fréchet–Urysohn space – Topological space
• Second-countable space – Topological space whose topology has a countable base
• Separable space – Topological space with a dense countable subset
• Sequential space – Topological space characterized by sequences

• "first axiom of countability", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics, Vol. 6 (Revised and completed ed.). Heldermann Verlag, Berlin. ISBN 3885380064.