In

topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a branch of

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a

**first-countable space** is a

topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

satisfying the "first

axiom of countabilityIn mathematics, an axiom of countability is a property of certain mathematical objects that requires the existence of a countable set with certain properties, while without it such sets might not exist....

". 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

sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

*U*_{1},

*U*_{2}, … of open neighbourhoods of

*x* such that for any open neighbourhood

*V* of

*x* there exists an integer

*i* with

*U*_{i} contained inIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

*V*.

## Examples and counterexamples

The majority of 'everyday' spaces in

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

are first-countable. In particular, every

metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

is first-countable. To see this, note that the set of open balls centered at

*x* with radius 1/

*n* for integers

*n* > 0 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 lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

).

Another counterexample is the ordinal space ω

_{1}+1 = [0,ω

_{1}] where ω

_{1} is the

first uncountable ordinalIn mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals...

number. The element ω

_{1} is a

limit pointIn mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...

of the subset

[0,ω

_{1}) even though no sequence of elements in

[0,ω

_{1}) has the element ω

_{1} as its limit. In particular, the point ω

_{1} in the space ω

_{1}+1 = [0,ω

_{1}] does not have a countable local base. The subspace ω

_{1} =

[0,ω

_{1}) is first-countable however, since ω

_{1} is the only such point.

The

quotient spaceIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

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 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.

## Properties

One of the most important properties of first-countable spaces is that given a subset

*A*, a point

*x* lies in the

closureIn mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

of

*A* if and only if there exists a

sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

{

*x*_{n}} in

*A* which

convergesThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

to

*x*. This has consequences for

limitsIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

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} →

*x*, where

*x*_{n} ≠

*x* for all

*n*, we have

*f*(

*x*_{n}) →

*L*. Also, if

*f* is a function on a first-countable space, then

*f* is continuous if and only if whenever

*x*_{n} →

*x*, then

*f*(

*x*_{n}) →

*f*(

*x*).

In first-countable spaces,

sequential compactnessIn mathematics, a topological space is sequentially compact if every sequence has a convergent subsequence. For general topological spaces, the notions of compactness and sequential compactness are not equivalent; they are, however, equivalent for metric spaces....

and

countable compactnessIn mathematics a topological space is countably compact if every countable open cover has a finite subcover.-Examples and Properties:A compact space is countably compact...

are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the

ordinal spaceIn mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...

[0,ω

_{1}). Every first-countable space is

compactly generatedIn topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:Equivalently, one can replace closed with open in this definition...

.

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.