Hubbry Logo
First-countable spaceFirst-countable spaceMain
Open search
First-countable space
Community hub
First-countable space
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
First-countable space
First-countable space
First-countable space
View on Wikipedia
from Wikipedia

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in 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

[edit]

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 with radius for integers form a countable local base at

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 where is the first uncountable ordinal number. The element is a limit point of the subset even though no sequence of elements in has the element as its limit. In particular, the point in the space does not have a countable local base. Since is the only such point, however, the subspace is first-countable.

The quotient space where the natural numbers on the real line are identified as a single point is not first countable.[1] However, this space has the property that for any subset and every element in the closure of there is a sequence in converging to 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.

Properties

[edit]

One of the most important properties of first-countable spaces is that given a subset a point lies in the closure of if and only if there exists a sequence in that converges to (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 is a function on a first-countable space, then has a limit at the point if and only if for every sequence where for all we have Also, if is a function on a first-countable space, then is continuous if and only if whenever then

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 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.

See also

[edit]

References

[edit]

Bibliography

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

First-countable space

View on Grokipedia
from Grokipedia
In , a first-countable space is a satisfying the , meaning that for every point xx in the space, there exists a countable collection Bx\mathcal{B}_x of open neighborhoods of xx such that every open neighborhood of xx contains some member of Bx\mathcal{B}_x. This local basis property distinguishes first-countable spaces from more general s, where neighborhood bases may be uncountable at certain points. The first countability axiom plays a crucial role in the study of convergence and continuity. In such spaces, a point lies in the closure of a set it is the from that set, providing a sequential of topological concepts that simplifies proofs and applications. Moreover, continuous functions between first-countable spaces map convergent sequences to convergent sequences, and the converse holds as well. Many familiar topological spaces are first-countable, including all metric spaces—where the open balls B(x,1/n)B(x, 1/n) for n∈Nn \in \mathbb{N} form a countable local basis at each point—and second-countable spaces, which have a countable basis for the entire and thus inherit first countability at every point. However, first countability is weaker than second countability. Subspaces and countable products of first-countable spaces remain first-countable, preserving the property under these operations. This axiom is foundational in , enabling the development of theorems on , separation, and metrizability in spaces with "tame" local structures.

Core Concepts

Definition

A XX is first-countable if every point x∈Xx \in X has a countable local basis, meaning there exists a countable collection {Bn(x)}n=1∞\{B_n(x)\}_{n=1}^\infty of open neighborhoods of xx such that for every open neighborhood UU of xx, there is some n∈Nn \in \mathbb{N} with Bn(x)⊆UB_n(x) \subseteq U. This local basis property serves as the core axiom of first-countability, ensuring that the topology around each point can be "generated" by a countable family of neighborhoods, which restricts the complexity of the space's local structure compared to spaces requiring uncountable bases. The countability condition is crucial, as it allows for the use of sequences to capture local topological behavior, distinguishing first-countable spaces from more general ones. The concept was formalized in the early as part of axiomatic topology by mathematicians like .

Countable Local Basis

In a first-countable , a countable local basis at a point xx consists of a countable collection {Bn(x)∣n∈N}\{B_n(x) \mid n \in \mathbb{N}\} of open neighborhoods of xx such that for every UU containing xx, there exists some nn with Bn(x)⊆UB_n(x) \subseteq U. This collection forms a fundamental system for the neighborhood filter at xx, meaning it generates all neighborhoods through finite intersections and ensures local topological properties can be analyzed using countably many sets. Often, such a basis can be chosen to be nested, satisfying B1(x)⊇B2(x)⊇⋯B_1(x) \supseteq B_2(x) \supseteq \cdots, though this nesting is not required by the ; the intersection ⋂n=1∞Bn(x)\bigcap_{n=1}^\infty B_n(x) frequently equals {x}\{x\} in Hausdorff spaces but may be larger otherwise. The nested form facilitates sequential approximations and convergence studies at xx, as it provides a decreasing chain of neighborhoods shrinking toward the point. To construct a countable local basis at xx, start with the neighborhood filter U(x)\mathcal{U}(x), which comprises all open sets containing xx; since the space is first-countable, this filter admits a countable fundamental system {Vn(x)∣n∈N}\{V_n(x) \mid n \in \mathbb{N}\}. A nested version can then be built by defining B1(x)=V1(x)B_1(x) = V_1(x) and Bk+1(x)=Vk+1(x)∩Bk(x)B_{k+1}(x) = V_{k+1}(x) \cap B_k(x) for k≥1k \geq 1, ensuring the resulting {Bn(x)}\{B_n(x)\} remains a local basis because intersections preserve the inclusion property for arbitrary neighborhoods. This method leverages the countable nature of the original system to produce a refined, nested basis without introducing uncountably many sets. Any two countable local bases at xx are equivalent up to refinement: for bases B={Bn(x)}\mathcal{B} = \{B_n(x)\} and C={Cm(x)}\mathcal{C} = \{C_m(x)\}, each Bn(x)B_n(x) contains some Cm(x)C_m(x) and vice versa, allowing one to be refined into the other by selective intersections. This equivalence underscores the structural flexibility of local bases while preserving their role in characterizing the near xx.

Illustrations

Examples

Euclidean spaces Rn\mathbb{R}^n equipped with the standard provide a fundamental example of first-countable spaces. For any point x∈Rnx \in \mathbb{R}^n, the collection of open balls B(x,1/n)B(x, 1/n) for n∈Nn \in \mathbb{N} forms a countable local basis at xx, as these balls decrease in radius and their intersections generate all neighborhoods of xx. More generally, every is first-countable. In a (X,d)(X, d), for each point x∈Xx \in X, the open balls B(x,1/n)B(x, 1/n) with n∈Nn \in \mathbb{N} constitute a countable local basis, since any open neighborhood of xx contains some ball of sufficiently small radius. The discrete topology on a countable set, such as the natural numbers N\mathbb{N}, is first-countable. Here, every singleton {x}\{x\} is an open neighborhood of xx, and the collection consisting of just {x}\{x\} itself serves as a countable (in fact, finite) local basis at xx. Topological manifolds and CW-complexes also exemplify first-countable spaces due to their local Euclidean structure. In a , each point has a neighborhood homeomorphic to an open subset of Rn\mathbb{R}^n, which inherits a countable local basis from the Euclidean topology; similarly, CW-complexes possess a cell decomposition that ensures local neighborhoods admit countable bases.

Counterexamples

A prominent example of a space that fails to be first-countable is an uncountable set XX equipped with the cofinite topology, where the open sets are the empty set and all subsets of XX with finite complements. In this topology, every nonempty open set is dense, but the space lacks a countable local basis at any point x∈Xx \in X. To see this, suppose for contradiction that there exists a countable collection {Un}n∈N\{U_n\}_{n \in \mathbb{N}} of open neighborhoods of xx forming a local basis at xx. Each UnU_n has finite complement, so the union ⋃n(X∖Un)\bigcup_{n} (X \setminus U_n) is countable. Since XX is uncountable, there exists some y∈X∖{x}y \in X \setminus \{x\} such that y∈⋂nUny \in \bigcap_{n} U_n. However, the cofinite set V=X∖{y}V = X \setminus \{y\} is an open neighborhood of xx, yet no UnU_n can be contained in VV because each UnU_n contains yy. This contradiction shows that no such countable local basis exists, implying that the cofinite topology on an uncountable set prevents the "shrinking" of neighborhoods to isolate points locally in a countable manner. Another illustrative counterexample is the cocountable topology on an uncountable set XX, defined by declaring a subset open if it is empty or its complement is at most countable. This topology also fails to be first-countable at every point x∈Xx \in X. Assume toward a contradiction that {Un}n∈N\{U_n\}_{n \in \mathbb{N}} is a countable local basis at xx. Each UnU_n has countable complement, so ⋃n(X∖Un)\bigcup_{n} (X \setminus U_n) remains countable. As XX is uncountable, pick y∈X∖{x}y \in X \setminus \{x\} outside this union, ensuring y∈⋂nUny \in \bigcap_{n} U_n. The set W=X∖{y}W = X \setminus \{y\} is then an open neighborhood of xx, but again, every UnU_n contains yy and thus cannot be subsets of WW. This failure demonstrates that countable collections of open sets cannot generate all necessary smaller neighborhoods around xx, leading to pathological local behavior where points cannot be separated countably from the rest of the space. A third classic counterexample arises in ordinal spaces, specifically the compact [0,ω1][0, \omega_1] consisting of all ordinals up to and including the first uncountable ordinal ω1\omega_1, endowed with the . This space is first-countable at every point α<ω1\alpha < \omega_1 due to the countable nature of the predecessors, but it fails first-countability at the endpoint ω1\omega_1. The basic open neighborhoods of ω1\omega_1 are of the form (α,ω1](\alpha, \omega_1] for α<ω1\alpha < \omega_1, forming an uncountable family. Suppose {Vn}n∈N\{V_n\}_{n \in \mathbb{N}} is a purported countable local basis at ω1\omega_1; each VnV_n contains some (αn,ω1](\alpha_n, \omega_1] with αn<ω1\alpha_n < \omega_1. Let α=sup⁡nαn<ω1\alpha = \sup_n \alpha_n < \omega_1, since the supremum of countably many countable ordinals is countable. Consider the open interval U=(α+1,ω1]U = (\alpha + 1, \omega_1], an open neighborhood of ω1\omega_1. For each nn, αn≤α<α+1\alpha_n \leq \alpha < \alpha + 1, so (αn,ω1](\alpha_n, \omega_1] contains α+1\alpha + 1 (as αn<α+1≤ω1\alpha_n < \alpha + 1 \leq \omega_1), hence α+1∈Vn\alpha + 1 \in V_n. But α+1∉U\alpha + 1 \notin U, so Vn⊄UV_n \not\subset U. Thus, no VnV_n is contained in UU, contradicting the local basis property. The character (minimal cardinality of a local basis) at ω1\omega_1 is ω1\omega_1, uncountable. This structural rigidity highlights how transfinite induction prevents countable isolation of the limit point, affecting properties like sequential compactness without metrizability.

Key Properties

Sequential Continuity

In first-countable topological spaces, the countable local basis at each point enables sequences to serve as a primary tool for characterizing continuity of functions. A fundamental result is that if XX is a first-countable space and YY is a Hausdorff topological space, then a function f:X→Yf: X \to Y is continuous at a point x∈Xx \in X if and only if, for every sequence (xn)(x_n) in XX converging to xx, the image sequence (f(xn))(f(x_n)) converges to f(x)f(x) in YY. To see this, first suppose ff is continuous at xx. Let VV be any neighborhood of f(x)f(x) in YY. Then f−1(V)f^{-1}(V) is a neighborhood of xx in XX, so there exists N∈NN \in \mathbb{N} such that xn∈f−1(V)x_n \in f^{-1}(V) for all n>Nn > N, implying f(xn)∈Vf(x_n) \in V for n>Nn > N. Thus, (f(xn))(f(x_n)) converges to f(x)f(x). For the converse, assume ff preserves sequential convergence at xx but is not continuous there. Then there exists a neighborhood VV of f(x)f(x) such that f(U)⊈Vf(U) \not\subseteq V for every neighborhood UU of xx. Let {Bk}k∈N\{B_k\}_{k \in \mathbb{N}} be a countable local basis at xx with Bk+1⊆BkB_{k+1} \subseteq B_k for all kk. For each kk, select xk∈Bk∖f−1(V)x_k \in B_k \setminus f^{-1}(V). The sequence (xk)(x_k) converges to xx because the BkB_k shrink to xx, but (f(xk))(f(x_k)) stays outside VV infinitely often and thus fails to converge to f(x)f(x) (since YY is Hausdorff), yielding a contradiction. This sequential criterion extends to the analysis of limits in first-countable spaces. Pointwise limits of continuous functions admit sequential characterizations of their behavior at points, leveraging the equivalence above to study convergence properties without full continuity. Additionally, while nets (generalizations of sequences indexed by directed sets) are required in general topologies to capture limits and continuity, first-countability allows reduction to sequences locally: a net in XX converges to xx if and only if it has a subnet (or cofinal subsequence) that is a sequence converging to xx. By contrast, non-first-countable spaces lack this equivalence, as sequential continuity need not imply topological continuity; nets become essential for precise characterizations.

Compactness Implications

In a first-countable , every compact is sequentially compact, meaning that every in the has a convergent (with limit in the ). This result follows from the structure of countable bases at each point, which allow the identification of cluster points of and the extraction of convergent subsequences within compact sets, where the finite subcover property ensures non-emptiness of limit point sets. More broadly, in first-countable spaces, sequential is equivalent to countable , the property that every countable open cover admits a finite subcover. Since implies countable , the aforementioned aligns with this equivalence, highlighting how first-countability bridges cover-based and sequence-based notions. The countable local basis at each point facilitates local refinement of covers: for any open cover of a compact , neighborhoods from the local bases can be used to extract finite subcovers around accumulation points, reinforcing the sequential characterization. However, compactness does not imply first-countability. A classic counterexample is the product space [0,1][0,1][0,1]^{[0,1]}, equipped with the , which is compact by . Yet, this space fails to be first-countable; for instance, the constant function f≡0f \equiv 0 (viewed as an element of the product) lacks a countable local basis, as any countable collection of neighborhoods cannot separate it from functions that differ on uncountably many coordinates. This illustrates that while first-countability strengthens compactness toward sequential properties, it is not a necessary condition for compactness itself.

Relations to Other Topological Notions

Comparison with Second-Countability

A is a whose topology admits a countable basis, meaning there exists a countable collection of open sets such that every open set in the topology can be expressed as a union of sets from this collection. In contrast, first-countability is a local property requiring only that each point has a countable local basis. Every is first-countable, as the countable global basis restricts to a countable collection of open neighborhoods at any fixed point, forming a local basis there. The converse does not hold: there exist first-countable spaces that are not second-countable. A standard example is the real line equipped with the , where every subset is open. In this space, the singleton set containing each point serves as a countable local basis at that point, satisfying first-countability. However, any basis for the entire must include all singletons, which form an uncountable collection, so the space fails to be second-countable. Second-countability also implies separability, the property of having a countable dense subset, since one can select a point from each nonempty basis element to form such a subset. First-countability does not imply separability, as the uncountable discrete space above has no countable dense subset—any dense set must intersect every singleton and thus be uncountable. A key connection between first-countability and metrizability arises in the Urysohn metrization theorem, which states that every regular Hausdorff is metrizable. However, first-countability is a strictly weaker local property than second-countability, and combining it with regularity and the Hausdorff axiom does not suffice for metrizability. The long line provides a classic counterexample: it is a first-countable, regular, Hausdorff manifold that is not second-countable. It is not metrizable; for example, it is countably compact but not compact. The Bing metrization theorem offers a that incorporates first-countability through the notion of Moore spaces, which are first-countable spaces admitting a development—a special type of open cover refining sequence. Specifically, the theorem asserts that a regular Hausdorff Moore space is metrizable it is collectionwise normal, meaning every discrete collection of closed sets can be separated by disjoint open sets. This result highlights how first-countability, when paired with a development and collectionwise normality, guarantees the existence of a compatible metric, extending earlier metrization criteria beyond global countability assumptions. The Nagata–Smirnov metrization theorem complements these ideas by characterizing metrizable spaces as regular Hausdorff spaces with a σ-locally finite basis, but its direct tie to first-countability emerges in contexts like paracompact spaces, where local countability aids in constructing such bases. Post-1970 developments further explored these links, particularly through the normal Moore space , which posits that every normal Moore space (hence first-countable and normal) is metrizable. Although the remains independent of ZFC, counterexamples constructed using forcing techniques in the demonstrated non-metrizable normal first-countable developable spaces under certain axioms, underscoring the subtle role of additional separation like collectionwise normality in ensuring metrizability. These advances refined the conditions under which first-countability contributes to global metric structures, especially in paracompact settings where local metrizability aligns with first-countability to imply full metrizability via extensions of Smirnov's theorem.
Add your contribution
Related Hubs
User Avatar
No comments yet.