First-countable Space · aiwiki.page

A first-countable space is a topological space in which each point has a countable local base of neighborhoods. This means that, at any point, a countable collection of neighborhoods suffices to refine every neighborhood of that point. First countability is a local condition: the collection may differ from point to point. Its principal importance in topology is that sequences can detect closure and test continuity, as they do in metric spaces. (math.utoronto.ca)

Definition and equivalent formulation

Let $X$ be a topological space and $x\in X$ . A countable local base at $x$ is a collection

\mathcal B_x=\{U_1,U_2,\ldots\}

of neighborhoods of $x$ such that, for every neighborhood $V$ of $x$ , some $U_n$ satisfies

x\in U_n\subseteq V.

The space $X$ is first-countable if such a collection exists at every point. A finite local base is allowed; its members can be repeated to obtain an indexed sequence. The base can always be chosen to consist of open sets. (math.utoronto.ca)

A useful equivalent formulation requires a decreasing local base. Given a countable open local base, put

W_n=\bigcap_{k=1}^{n}U_k.

Then each $W_n$ is an open neighborhood of $x$ , the collection remains a local base, and

W_{n+1}\subseteq W_n.

Consequently, choosing $x_n\in W_n$ guarantees that $x_n$ converges to $x$ : every neighborhood contains all sufficiently late terms. (math.utoronto.ca)

Examples and relation to other countability conditions

Every metric space is first-countable. If $d$ is its metric, the open balls

\{B_d(x,1/n):n\geq1\}

form a local base at $x$ . Given a neighborhood containing $B_d(x,\varepsilon)$ , choose $n$ with $1/n<\varepsilon$ . In particular, Euclidean spaces are first-countable. (math.uchicago.edu)

Every second-countable space is also first-countable. A countable base for the whole topology gives a countable local base at $x$ by retaining its members that contain $x$ . The converse fails: an uncountable discrete space is first-countable because $\{\{x\}\}$ is a local base at each point, but no countable global base can contain all its singletons. Thus first countability does not require the underlying set to be countable or the topology to have a countable base. (math.toronto.edu)

A standard nonmetrizable example is the Sorgenfrey line: the real line with the topology generated by intervals $[a,b)$ . At $x$ , the intervals

[x,x+1/n),\qquad n\geq1,

form a countable local base. This space is separable but not second-countable. Since every separable metrizable space is second-countable, it is not metrizable. First countability is therefore strictly weaker than metrizability, even for Hausdorff spaces. (math.wustl.edu)

Closure and sequences

The central sequence theorem states that, for a first-countable space $X$ , a subset $A\subseteq X$ , and $x\in X$ ,

x\in\overline A \quad\Longleftrightarrow\quad \text{some sequence }(a_n)\text{ in }A\text{ converges to }x,

where $\overline A$ is the closure of $A$ . (sites.math.northwestern.edu)

For the forward direction, take a decreasing local base $(W_n)$ at $x$ . Every $W_n$ intersects $A$ , so choose

a_n\in A\cap W_n.

The local-base property gives $a_n\to x$ . Conversely, if a sequence in $A$ converges to $x$ , every neighborhood of $x$ intersects $A$ , placing $x$ in its closure. This converse holds in every topological space. (sites.math.northwestern.edu)

It follows that a subset of a first-countable space is closed exactly when it contains every limit of every convergent sequence of its points. Repeated terms are permitted: if $x\in A$ , the constant sequence at $x$ already witnesses membership in $\overline A$ . No separation assumption is needed for these statements. (math.utoronto.ca)

Continuity and weaker sequence conditions

If $X$ is first-countable and $Y$ is any topological space, a function $f:X\to Y$ is a continuous function if and only if

x_n\to x\text{ in }X \quad\Longrightarrow\quad f(x_n)\to f(x)\text{ in }Y.

Only the domain needs to be first-countable. Continuous functions preserve convergent sequences without any countability assumption; first countability supplies the reverse implication. (sites.math.northwestern.edu)

First countability is sufficient, but not necessary, for sequences to determine topological information. Every first-countable space is a Fréchet–Urysohn space, meaning that each point in a set’s closure is the limit of a sequence from that set. Every Fréchet–Urysohn space is a sequential space, meaning that sequentially closed subsets are closed. Neither converse holds in general. These distinctions separate the existence of countable local bases from the weaker ability of sequences to detect closure or closedness. (arxiv.org)

Subspaces, products, and images

First countability is preserved by taking subspaces: if $S\subseteq X$ and $x\in S$ , intersect a countable local base at $x$ with $S$ . It is also invariant under homeomorphisms. (arxiv.org)

Finite and countable products of first-countable spaces are first-countable in the product topology. At a point of a countable product, use basic neighborhoods restricting finitely many coordinates, with each restriction chosen from the corresponding countable local base. There are only countably many such choices. (math.toronto.edu)

Uncountable products need not be first-countable. For example, $\{0,1\}^{I}$ , with discrete factors and uncountable $I$ , is not first-countable. A proposed countable local base would involve only countably many coordinates after choosing a basic product neighborhood inside each member. A neighborhood restricting a coordinate outside that countable collection could not contain any of those basic neighborhoods. (math.toronto.edu)

Arbitrary continuous images need not preserve first countability. Open continuous surjections do preserve it: the images of a countable local base at a preimage point form a local base at its image. (arxiv.org)

A space where sequences miss closure

Let $X$ be uncountable with the cocountable topology: its nonempty open sets are those whose complements are countable. This space is not first-countable. If $(U_n)$ were a local base at $x$ , the union of their countable complements would be countable. Choose

y\ne x,\qquad y\in\bigcap_n U_n.

Then $X\setminus\{y\}$ is a neighborhood of $x$ containing none of the proposed base members, a contradiction. (math.toronto.edu)

Here every sequence converging to $x$ must eventually equal $x$ : remove the countable set of its values other than $x$ to obtain a neighborhood of $x$ . Yet every uncountable subset is dense. Thus, if $A$ is uncountable and $x\notin A$ , then $x\in\overline A$ , although no sequence in $A$ converges to $x$ . (arxiv.org)

In general spaces, nets and filters provide convergence descriptions of closure without a first-countability assumption. First countability is precisely a useful local-base hypothesis under which ordinary sequences already suffice for this purpose. It does not, by itself, ensure unique limits; uniqueness is guaranteed in a Hausdorff space. (math.uchicago.edu)

References

Sequences, weak T-axioms, and first countabilitymath.utoronto.ca
Math 344-1: Introduction to Topologysites.math.northwestern.edu
Topology - Imath.uchicago.edu
MAT327 Big Listmath.toronto.edu
Chapter 3 Top Spacesmath.wustl.edu
Countabilitymath.toronto.edu
Sequential convergence in topological spacesarxiv.org
Arbitrary productsmath.toronto.edu