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 $$N_1, N_2, \ldots$$ 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 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 $$\omega_1 + 1 = \left[0, \omega_1\right]$$ 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 $$\R / \N$$ 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 $$\left(x_n\right)_{n=1}^{\infty}$$ in $$A$$ which 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 which 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.