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First-countable space

Topological space where each point has a countable neighbourhood basis

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Source updated May 23, 2026

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.

01Examples 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 2^{-n} for all natural numbers n 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. 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.

02Properties

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.

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Sources and credits

This article is adapted from the Wikipedia article “First-countable space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Fréchet-Urysohn space

In the field of topology, a Fréchet-Urysohn space is a topological space X {\displaystyle X} with the property that for every subset S ⊆ X {\displaystyle S\subseteq X} the closure of S {\displaystyle S} in X {\displaystyle X} is identical to the sequential closure of S {\displaystyle S} in X . {\displaystyle X.} Fréchet-Urysohn spaces are a special type of sequential space.

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T {\displaystyle T} is second-countable if there exists some countable collection U = { U i } i = 1 ∞ {\displaystyle {\mathcal {U}}=\{U_{i}\}_{i=1}^{\infty }} of open subsets of T {\displaystyle T} such that any open subset of T {\displaystyle T} can be written as a union of elements of some subfamily of U {\displaystyle {\mathcal {U}}} .

Separable space

In mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence n = 1 ∞ {\displaystyle (x_{n})_{n=1}^{\infty }} of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense.