Regular space
In topology, a regular space is a Hausdorff topological space in which open sets can separate any point from any closed set not containing it; specifically, for every closed subset F of the space X and every point x \notin F, there exist disjoint open sets U_x and V_x such that x \in U_x and F \subseteq V_x.[1] This separation axiom, often denoted as T_3 when combined with the Hausdorff condition, ensures a level of "niceness" in the topology that allows for stronger continuity properties and embedding results.[2] Regular spaces form a key part of the hierarchy of separation axioms, positioned strictly between Hausdorff (T_2) spaces and normal (T_4) spaces, where the latter requires disjoint closed sets to be separable by disjoint open sets.[2] Equivalent formulations include the property that every open neighborhood of a point contains a closed neighborhood of that point.[2] Notably, regularity is preserved under subspaces and arbitrary products, making it a stable property for constructing more complex spaces from simpler ones.[1] However, not all regular spaces are normal; counterexamples exist, though every second-countable regular space (e.g., with a countable basis) is normal.[2] Examples of regular spaces abound in familiar settings: all metric spaces are regular (in fact, normal), including Euclidean spaces \mathbb{R}^n with the standard topology and any manifold, as their local Euclidean structure induces a metric topology.[3] The real line \mathbb{R} under the usual topology is regular, as balls around points and complements of closed sets can be separated by disjoint intervals.[1] Regularity plays a crucial role in advanced theorems, such as Urysohn's lemma, which guarantees continuous real-valued functions separating points from closed sets in normal spaces (extending to regular contexts), and the Urysohn metrization theorem, stating that every second-countable, regular Hausdorff space is metrizable.[1] These properties underpin much of geometric topology and analysis, ensuring spaces support continuous extensions and approximations essential for studying embeddings and compactness.[2]Definitions and Terminology
Formal Definition
In topology, a regular space is defined as follows: Let (X, \tau) be a topological space, where \tau denotes the collection of open sets. The space X is regular if, for every point x \in X and every closed set C \subseteq X such that x \notin C, there exist disjoint open sets U, V \in \tau with x \in U and C \subseteq V.[4] An equivalent point-set formulation states that X is regular if and only if, for every closed set C \subseteq X and every point x \in X \setminus C, there exists an open neighborhood U \in \tau of x such that the closure \mathrm{cl}(U) (with respect to \tau) satisfies \mathrm{cl}(U) \cap C = \emptyset.[5] This separation condition constitutes the core regular axiom, often denoted R_2, and serves as the initial motivation for the T_3 separation axiom in the absence of the Hausdorff (T_2) assumption.[6]T3 Space and Related Terms
A T3 space is a topological space that is both regular and T1, meaning that for every point x not contained in a closed set F, there exist disjoint open sets U containing x and V containing F.[7] This separation property ensures that points can be distinguished from closed sets not containing them via open neighborhoods, building on the regular condition while incorporating the T1 axiom for finite set closures.[8] The terminology for T3 spaces exhibits variations across texts; some definitions include the T0 or T1 axiom instead of or alongside T2, but the standard modern usage specifies regular plus T2 (Hausdorff) to guarantee the implications hold without ambiguity, as regular + T1 implies T2.[9] To circumvent these inconsistencies, the phrase "regular Hausdorff space" is often preferred in contemporary literature, emphasizing the combined properties explicitly.[6] The notion of T3 spaces traces its origins to early 20th-century developments in topology, with regular spaces first defined by Vietoris in 1921 and the T-notation for separation axioms formalized by Alexandroff and Hopf in their 1935 textbook to resolve terminological ambiguities.[10] Mathematicians like Kuratowski contributed related foundational work on closure operators in 1922, influencing the precise articulation of these axioms amid evolving understandings of separation in topological structures.[10] In contrast, preregular spaces weaken the Hausdorff condition, requiring only that topologically distinguishable points—those not sharing all neighborhoods—can be separated by disjoint open sets, whereas in T3 spaces, all distinct points are separable regardless.[11]Separation Axioms Framework
Comparisons with Other Axioms
A regular space, as defined in this article (a Hausdorff space satisfying the point-closed set separation axiom, equivalent to the T3 axiom), implies the preregular axiom (also denoted R1 or T_{3.5}), which requires that any two topologically distinguishable points can be separated by disjoint open neighborhoods.[12] Since regular spaces are Hausdorff (T2), they also imply the T1 and T0 axioms. As a separation property, regularity occupies a position between the Hausdorff axiom (T2, which it includes) and normality (T4). Specifically, a regular space is T3 by definition in this convention, implying T2, T1, and T0. Note that in some modern sources, "regular" refers only to the point-closed set separation without Hausdorff, but here it includes it to align with traditional usage where regular is synonymous with T3.[13] The key implications involving regularity and adjacent axioms can be summarized as follows:| From \ To | Preregular (R1) | T0 | T1 | T2 (Hausdorff) | T3 | T4 (Normal + T1) |
|---|---|---|---|---|---|---|
| Regular | Yes | Yes | Yes | Yes | Yes | No |