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In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X {\displaystyle X} is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x {\displaystyle x} in X {\displaystyle X} there exists a sequence N 1 , N 2 , … … {\displaystyle N_{1},N_{2},\ldots } of neighbourhoods of x {\displaystyle x} such that for any neighbourhood N {\displaystyle N} of x {\displaystyle x} there exists an integer i {\displaystyle i} with N i {\displaystyle N_{i}} contained in N . {\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

• Properties

• See also

• References

• Bibliography

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

The quotient space R / N {\displaystyle \mathbb {R} /\mathbb {N} } where the natural numbers on the real line are identified as a single point is not first countable.cite-ref-1[1] However, this space has the property that for any subset A {\displaystyle A} and every element x {\displaystyle x} in the closure of A , {\displaystyle A,} there is a sequence in A {\displaystyle A} converging to x . {\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 A , {\displaystyle A,} a point x {\displaystyle x} lies in the closure of A {\displaystyle A} if and only if there exists a sequence ( x n ) n = 1 ∞ ∞ {\displaystyle \left(x_{n}\right)_{n=1}^{\infty }} in A {\displaystyle A} that converges to x . {\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 f {\displaystyle f} is a function on a first-countable space, then f {\displaystyle f} has a limit L {\displaystyle L} at the point x {\displaystyle x} if and only if for every sequence x n → → x , {\displaystyle x_{n}\to x,} where x n ≠ ≠ x {\displaystyle x_{n}\neq x} for all n , {\displaystyle n,} we have f ( x n ) → → L . {\displaystyle f\left(x_{n}\right)\to L.} Also, if f {\displaystyle f} is a function on a first-countable space, then f {\displaystyle f} is continuous if and only if whenever x n → → x , {\displaystyle x_{n}\to x,} then f ( x n ) → → f ( x ) . {\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 [ 0 , ω ω 1 ) . {\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 – Property of 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

cite-note-11. ↑ (Engelking 1989, Example 1.6.18)

• "first axiom of countability", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

• citerefengelking1989Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics, Vol. 6 (Revised and completed ed.). Heldermann Verlag, Berlin. ISBN 3885380064.