Fact-checked by Grok 2 weeks ago

Axiom of countability

In topology, the countability axioms are properties that require the existence of countable collections related to the open sets of a topological space. The second axiom of countability is a fundamental such property: a topological space satisfies it if it admits a countable basis—a countable collection of open sets such that every open set in the topology can be expressed as a union of basis elements.^[1] This axiom ensures that the space has a "manageable" structure in terms of its open sets, distinguishing it from more general topological spaces that may require uncountably many basis elements.^[2] In contrast to the first axiom of countability, which requires only that every point in the space has a countable local basis of neighborhoods, the second axiom applies globally to the entire topology and is strictly stronger.^[3] Every second-countable space is first-countable, but the converse does not hold; for example, the Sorgenfrey line (the real line with the lower limit topology) is first-countable but not second-countable.^[1] Second countability implies several other desirable properties, including the Lindelöf condition (every open cover has a countable subcover) and separability (the existence of a countable dense subset).^[3] Classic examples of second-countable spaces include the real line \mathbb{R} with the standard topology, where the open intervals with rational endpoints form a countable basis, and more generally, any separable metric space.^[2] Subspaces and countable products of second-countable spaces remain second-countable, making this axiom particularly useful for studying Euclidean spaces and manifolds.^[1] In metrizable spaces, second countability is equivalent to separability and the Lindelöf property, highlighting its role in bridging topological and metric concepts.^[3]

Definitions

First axiom of countability

In topology, a topological space X satisfies the first axiom of countability, also known as being first countable, if every point in X has a countable local basis of open neighborhoods.^[4] This condition ensures that the neighborhood structure around each point can be "tamed" by a countable collection, allowing for more manageable local analysis compared to spaces without such restrictions.^[5] A local basis at a point x \in X, denoted \mathcal{B}_x, is a collection of open sets containing x such that for every open neighborhood U of x, there exists some V \in \mathcal{B}_x with x \in V \subseteq U.^[5] The first axiom requires that this \mathcal{B}_x is countable for each x, meaning it can be enumerated as a sequence or list without repetition beyond necessity. This local countability contrasts with the second axiom of countability, which imposes a countable basis on the entire topology globally.^[4] Formally, X is first countable if for each x \in X, there exists a countable family of open neighborhoods \{U_n(x)\}_{n \in \mathbb{N}} such that for any open set U containing x, there is some n with U_n(x) \subseteq U.^[5] This axiom was introduced by Felix Hausdorff in his foundational work on set-theoretic topology in 1914, where it served as a key local property to study convergence and separation in abstract spaces.^[4]

Second axiom of countability

A topological space X satisfies the second axiom of countability, or is second-countable, if its topology admits a countable basis.^[1] This means there exists a countable collection \mathcal{B} = \{B_n \mid n \in \mathbb{N}\} of open subsets of X such that every open set in the topology of X can be expressed as a union of elements from \mathcal{B}.^[1] Formally, the collection \mathcal{B} forms a basis if for every open set U \subseteq X and every point x \in U, there exists some B_n \in \mathcal{B} such that x \in B_n \subseteq U.^[1] This condition ensures that \mathcal{B} generates the entire topology through arbitrary unions, providing a countable "building block" structure for all open sets.^[1] Any second-countable space is also first-countable.^[6] To see this, fix a point x \in X; the subcollection \{B_n \in \mathcal{B} \mid x \in B_n\} is countable and serves as a local basis at x, since for any open neighborhood U of x, there exists B_n with x \in B_n \subseteq U.^[6] Thus, the global countability of \mathcal{B} yields countable local bases at each point via countable subcollections.^[6] In contrast to the first axiom of countability, which requires only a countable local basis at each individual point without a uniform global structure, the second axiom imposes a stronger condition by mandating a single countable basis for the whole space.^[1]

Equivalent formulations

Local bases in first countability

In a topological space X, the first axiom of countability is equivalently formulated by requiring that every point x \in X possesses a countable local basis, that is, a countable collection \{B_n(x)\}_{n \in \mathbb{N}} of open neighborhoods of x such that for any open neighborhood U of x, there exists some n with B_n(x) \subseteq U.^[7] This local basis can often be chosen to be nested, satisfying B_1(x) \supseteq B_2(x) \supseteq \cdots, which simplifies constructions in proofs.^[7] The presence of such a countable local basis at each point enables a sequential characterization of topological notions that typically require nets or filters in more general spaces. Specifically, in a first-countable space, convergence can be described using sequences rather than nets: a net converges to a point if and only if there is a cofinal subsequence that converges to it as a sequence.^[8] This simplification arises because the countable basis allows one to extract a sequence from any convergent net by selecting points in successively smaller basis elements. A key theorem in this context states that a topological space X is first-countable if and only if, for every subset A \subseteq X, the closure \overline{A} equals A union the set of all limits of convergent sequences in A.^[5] Equivalently, a point x lies in \overline{A} if and only if there exists a sequence \{x_n\} \subseteq A such that x_n \to x.^[8] Thus, a subset F \subseteq X is closed if and only if it contains all limits of convergent sequences from F.^[7] This sequential criterion for closed sets holds precisely due to the countable local bases, which facilitate the construction of sequences witnessing limit points. Such formulations are central in general topology, as explored in standard texts like Munkres' Topology.^[1] Second-countable spaces satisfy first countability, since a countable basis for the entire space restricts to a countable local basis at each point.^[5]

Countable bases in second countability

The second axiom of countability, or second-countability, admits an equivalent formulation in terms of subbases. A subbasis for a topological space X is a collection \mathcal{S} of open subsets whose finite intersections form a basis for the topology on X. A space satisfies the second axiom of countability if and only if it has a countable subbasis.^[9] To see this equivalence, note that any countable basis \mathcal{B} = \{B_n \mid n \in \mathbb{N}\} serves directly as a countable subbasis, as the collection of all finite intersections of elements from \mathcal{B} is countable and forms a basis for the topology, and \mathcal{B} generates the topology via arbitrary unions. Conversely, if \mathcal{S} = \{S_n \mid n \in \mathbb{N}\} is a countable subbasis, then the collection of all finite intersections \bigcap_{i=1}^k S_{n_i} (for k \in \mathbb{N}, n_i \in \mathbb{N}) is countable, as there are countably many finite subsets of \mathbb{N}, and this collection forms a basis for the topology.^[9] Second-countability also imposes restrictions on the cardinality of the topology itself. In a second-countable space with countable basis \mathcal{B}, every open set is a union of at most countably many elements from \mathcal{B}, so the collection of all possible such unions has cardinality at most $2^{\aleph_0}, the cardinality of the continuum. Thus, the topology has at most continuum many open sets.^[10]

Properties and implications

Implications between axioms

A second-countable topological space is always first-countable. To see this, let (X, \tau) be a second-countable space with countable basis \{B_n \mid n \in \mathbb{N}\}. For any point x \in X, the collection \{B_n \mid x \in B_n\} forms a countable local basis at x, since for any open neighborhood U of x, there exists some B_k such that x \in B_k \subset U, ensuring the collection satisfies the local basis property.^[11]^[12] The converse does not hold: first-countability does not imply second-countability. An uncountable set equipped with the discrete topology provides a counterexample, as the singleton sets form a countable local basis at each point, making the space first-countable, but any basis for the topology must include all singletons, which is uncountable.^[11]^[13] The countability axioms are defined for general topological spaces and hold independently of separation properties such as T1 or Hausdorffness, though they are frequently studied in regular or Hausdorff contexts to ensure desirable behaviors like metrizability.^[14]^[12] As a consequence, every second-countable space is Lindelöf, meaning that every open cover admits a countable subcover.^[15]^[16]

Connections to compactness and metrizability

A compact Hausdorff space is metrizable if and only if it is second-countable. This equivalence arises as a special case of Urysohn's metrization theorem, which states that every regular second-countable Hausdorff space is metrizable, combined with the fact that compact Hausdorff spaces are normal (and hence regular).^[17] In a compact second-countable space, the countable basis forms an open cover, and by compactness, there exists a finite subcollection of these basis elements that covers the entire space. This property underscores the interplay between countability and compactness, ensuring that such spaces are metrizable and exhibit controlled topological complexity. Second-countable spaces play a key role in dimension theory, particularly for finite-dimensional cases like topological manifolds, which are defined to be second-countable, Hausdorff, and locally Euclidean of dimension n, thereby possessing inductive dimension exactly n. While second-countable spaces can have infinite inductive dimension—for instance, the countable product \mathbb{R}^\mathbb{N} does so—the emphasis in applications often falls on finite-dimensional structures where second countability ensures manageable covering properties.^[18]^[19] In metric spaces, second countability is equivalent to separability (having a countable dense subset) and to the Lindelöf property (every open cover admits a countable subcover). Additionally, in first-countable spaces, compactness implies sequential compactness.^[20]^[21]

Examples

Spaces satisfying both axioms

Euclidean spaces provide a fundamental example of topological spaces satisfying both the first and second axioms of countability. The space \mathbb{R}^n equipped with the standard topology possesses a countable basis consisting of open rectangles formed by products of open intervals with rational endpoints, which establishes second countability; first countability follows as a consequence of second countability in general topological spaces.^[22] Separable metric spaces also satisfy both axioms. In a separable metric space, the existence of a countable dense subset allows the construction of a countable basis by taking open balls of rational radii centered at points of the dense subset, thereby proving second countability; metric spaces are inherently first countable due to the countable collection of balls of rational radii around each point.^[23] Finite-dimensional smooth manifolds are defined to be second countable. By construction, a smooth manifold is a second-countable Hausdorff space that is locally Euclidean, with the countable basis arising from the countable collection of chart domains; the local Euclidean structure ensures first countability at each point.^[24] The Hilbert cube, defined as the product \prod_{n=1}^\infty \left[0, \frac{1}{n}\right] with the product topology, is a compact metric space that is separable, hence second countable. Its first countability follows from the metric structure.^[25] Spaces satisfying both axioms, such as these examples, are Lindelöf, meaning every open cover admits a countable subcover.^[22]

Spaces satisfying one but not the other

While every second-countable space is first-countable, the converse does not hold, as there exist topological spaces that satisfy the first axiom of countability but fail the second. This implication follows from the fact that a countable basis for the entire topology can be used to construct a countable local basis at each point by selecting those basis elements containing the point and shrinking them appropriately using the continuity of the topology.^[7] A classic example of a first-countable space that is not second-countable is the discrete topology on an uncountable set X. In this topology, every subset is open, so the singleton \{x\} forms a countable (in fact, finite) local basis at each point x \in X, satisfying first countability. However, any basis for the topology must include all singletons to generate the open sets, requiring an uncountable collection, so the space lacks a countable basis.^[13] Another example is the Sorgenfrey line, which is the real line \mathbb{R} equipped with the lower limit topology generated by the half-open intervals [a, b) for a < b. At each point x \in \mathbb{R}, the collection \{[x, x + \frac{1}{n}) \mid n \in \mathbb{N}\} serves as a countable local basis, making the space first-countable. Yet, it has no countable basis for the entire topology, as any countable collection of basis elements can only "cover" countably many such intervals in a way that fails to generate all open sets, necessitating an uncountable basis.^[26] The Moore plane, also known as the Niemytzki plane, provides a further illustration: it consists of the closed upper half-plane \mathbb{R}^2_{\geq 0} where points above the x-axis have the usual Euclidean neighborhoods, while points on the x-axis have neighborhoods consisting of tangent open discs above the axis. Each point has a countable local basis—Euclidean balls for interior points and sequences of shrinking tangent discs for boundary points—ensuring first countability. Nevertheless, the space is not second-countable, as the boundary points require uncountably many distinct tangent disc bases to separate them properly.^[27] These examples are typically non-metrizable, highlighting how first countability alone does not guarantee the stronger uniformity of second countability.

Lindelöf property

The Lindelöf property is a covering axiom in topology stating that a topological space is Lindelöf if every open cover of the space admits a countable subcover. This condition ensures a form of "countable refinement" for open covers, weaker than compactness but stronger than certain other global properties in non-metrizable settings.^[15] Second-countable spaces, which possess a countable basis for their topology, are always Lindelöf, as any open cover can be refined to a countable subcollection from the basis. However, the converse does not hold: there exist Lindelöf spaces that lack a countable basis. A classic example is the Sorgenfrey line, the real line equipped with the lower limit topology generated by half-open intervals [a, b); this space is hereditarily Lindelöf and separable but not second-countable, as any countable collection of basis elements fails to generate all open sets.^[28] In the context of separation axioms, every regular Lindelöf space is normal, meaning disjoint closed sets can be separated by disjoint open sets.^[29] This result ties the Lindelöf property to countability through its interaction with regularity, and since second-countability implies both regularity preservation and Lindelöf, it strengthens such normality guarantees in countable-basis settings.^[29]

Separability

In topology, a topological space X is said to be separable if it contains a countable dense subset, meaning there exists a countable set D \subseteq X such that every non-empty open subset of X intersects D.^[30]^[31] Every second-countable topological space is separable. To see this, let \mathcal{B} = \{B_n : n \in \mathbb{N}\} be a countable basis for the topology on X. For each non-empty B_n, select a point x_n \in B_n. The set D = \{x_n : n \in \mathbb{N}\} is countable and dense in X, since any non-empty open set U \subseteq X contains some basis element B_k, and thus intersects D at x_k.^[32]^[31] In the context of metric spaces, separability and second countability are equivalent properties. Specifically, a metric space is separable if and only if it is second-countable. This equivalence highlights the close relationship between the existence of a countable dense subset and the existence of a countable basis in metric topologies.^[31] A classic example is the real line \mathbb{R} with the standard Euclidean topology, which is separable because the set of rational numbers \mathbb{Q} is countable and dense in \mathbb{R}; every non-empty open interval in \mathbb{R} contains rational points. Consequently, \mathbb{R} is also second-countable, with the collection of open intervals with rational endpoints forming a countable basis.^[30]

References

[1]
[PDF] Section 30. The Countability Axioms
Dec 8, 2016 · Definition. If a topological space X has a countable basis for its topology, then. X satisfies the Second Countability Axiom, or is second- ...
[2]
[PDF] 4. Countability
First, take care to note that the definition does not say that every basis of the topology is countable, just that there is at least one countable basis. In ...
[3]
[PDF] (1) The first countability axiom. (2) The second
We have studied four basic countability properties: (1) The first countability axiom. (2) The second countability axiom,. (3) The Lindel6f condition.
[4]
First axiom of countability - Encyclopedia of Mathematics
Feb 7, 2011 · A concept in set-theoretic topology. A topological space satisfies the first axiom of countability if the defining system of neighbourhoods ...
[5]
[PDF] 3 COUNTABILITY AND CONNECTEDNESS AXIOMS
X satisfies the first axiom of countability or is first countable iff for each x ∈ X, there is a countable neighbourhood basis at x. Any metrisable space ...
[6]
[PDF] Metric & Topological Spaces - alexwlchan
A topological space (X, τ) is called secound countable if it has a countable base of open sets. Clearly, second countable implies first countable. Consider: for ...
[7]
[PDF] Topology (H) Lecture 13
Apr 22, 2021 · In topology, an axiom of count- ability is a topological property that asserts the existence of a countable set with certain properties. There ...
[8]
[PDF] 5. Sequences, weak T-axioms, and first countability
Definition 5.3. A topological space (X,Τ ) is said to be first countable if every point in X has a countable local basis. This is the key property that allows ...
[9]
[PDF] Class Notes for Math 871: General Topology, Instructor Jamie ...
If X has a countable subbasis, then X is 2nd countable. Proof. If S is a countable subbasis then B = {S1 ∩ททท∩ Sn|n ≥ 0,Si ∈ S} is a countable basis.
[10]
[PDF] DESCRIPTIVE SET THEORY PROBLEM SET 1. Let X be a second ...
Let X be a second-countable topological space. (a) Show that X has at most continuum many open subsets. (b) Let α,β,γ denote ordinals.
[11]
[PDF] Review of point-set topology Andrew Putman - Academic Web
A space X is second countable if there is a countable basis for its topology. It is clear that all second countable spaces are first countable. It is not ...<|separator|>
[12]
[PDF] INTRODUCTION TO TOPOLOGY Contents 1. Basic concepts 1 2 ...
A topological space (X, τ) is called first countable, if every point x ∈ X has a countable neighbourhood basis. The topological space (X, τ) is second countable ...
[13]
[PDF] 1 Hausdorff spaces - Math 535 - General Topology Additional notes
Sep 14, 2012 · For example, consider X an uncountable set endowed with the discrete topology. Then X is first-countable but not second-countable. Remark 2.13.
[14]
[PDF] Chapter 4: Topological Spaces - UC Davis Mathematics
The discrete topology is first countable, and if X is countable, then it is second countable. It is often useful to define a topology in terms of a base. T h ...
[15]
Lindelöf space - PlanetMath
Mar 22, 2013 · A second-countable space is Lindelöf. A compact space is Lindelöf. A regular (http://planetmath.org/T3Space) ...
[16]
Second-countable implies Lindelof - Topospaces
Jul 20, 2008 · This article gives the statement and possibly, proof, of an implication relation between two topological space properties.
[17]
[PDF] Normal Spaces, Regular Spaces, Urysohn metrization - UTK Math
Urysohn metrization theorem. Let X be Hausdorff and second-countable. Then X is metrizable if and only if X is normal. The following lemma used in the proof ...
[18]
[PDF] Chapter 1: Smooth manifolds - Arizona Math
Sep 2, 2010 · A topological manifold is Hausdorff, second countable, and locally Euclidean of dimension n. A manifold of dimension n is also called an n- ...
[19]
Sets of infinite Hausdorff dimension in a second countable metric ...
Jul 18, 2020 · The infinite power RN is separable metrizable (hence second countable) and has infinite inductive dimension. This is a topologically defined ...finite dimensional compact Hausdorff space, not second countableIs second-countability a topological invariant? - Math Stack ExchangeMore results from math.stackexchange.com
[20]
Separability of a Metric Space Is Equivalent to the Existence of a ...
Then the following conditions are equivalent: X is separable;. X is second countable, i.e., it has a countable open base;. X is Lindelöf ...
[21]
[PDF] Compactness, countability, function spaces: examples - UTK Math
If X is locally compact, σ-compact (for instance, a topological manifold), then Cc(X, M) is second-countable if M is separable metric. For each i ≥ 1, let Bi be ...
[22]
[PDF] Lecture Notes on Topology for MAT3500/4500 following J. R. ... - UiO
Nov 29, 2010 · A space having a countable basis at each of its points is said to satisfy the first countability axiom, or to be first-countable. Every ...
[23]
[PDF] Section 9.6. Separable Metric Spaces
Dec 10, 2022 · A metric space X is separable provided there is a countable subset of X that is dense in X. Note 9.6.A. By the definition of “point of closure,” ...
[24]
https://edu.fjfi.cvut.cz/studijni-materialy/Ing/4.%20ro%C4%8Dn%C3%ADk/GMF2/John%20M.%20Lee%20(auth.)%20-%20Introduction%20to%20Smooth%20Manifolds-Springer%20New%20York%20(2003).pdf
[25]
Hilbert Cube - an overview | ScienceDirect Topics
H ω is a separable compact metric space, hence a second countable compactum and a Polish space. H ω is called the Hilbert cube. It is a compact subset of the ...
[26]
[PDF] Chapter III Topological Spaces
b) Use part a) to prove that the Sorgenfrey line is not second countable. (Hint: Show that otherwise there would be a countable base of sets of the form but.
[27]
[PDF] MATH 561 - HOMEWORK 2
Thus every point in the Moore plane has a countable local basis, and so the Moore plane is first-countable. However, the Moore plane is not second-countable.
[28]
[PDF] 10. Covering properties - TU Graz
Observe that the Sorgenfrey line is hereditarily Lindelöf but not second countable. For metric spaces, however, we have the following important result. Theorem.
[29]
[PDF] A Comparison of Lindelof-type Covering Properties of Topological ...
2.1 Definition. A topological space X is called Lindelöf if every open cover of X has a countable subcover. 2.2 Examples. The following are straightforward ...
[30]
Separable Space -- from Wolfram MathWorld
A topological space having a countable dense subset. An example is the Euclidean space R^n with the Euclidean topology.
[31]
separable space - PlanetMath.org
Mar 22, 2013 · All second-countable spaces are separable. A metric space is separable if and only if it is second-countable.
[32]
every second countable space is separable - PlanetMath
Mar 22, 2013 · Let X X be a second countable space and let B ℬ be a countable base. For every non-empty set B B in B ℬ , choose a point xB∈B x B ∈ B . The ...