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A space is regular if for every closed set F and x ∉ F, there exist disjoint open sets separating them. T₃ = regular + T₀. Regular spaces separate points from closed sets; metric spaces are regular. Products of regular spaces are regular; regularity is preserved under continuous images (unlike T₂).
A topological space X is regular if, given any closed set F and any point x not in F, there exist disjoint open sets U and V with x ∈ U and F ⊆ V. In other words, a point and a closed set that does not contain it can always be "separated" by open neighborhoods that do not overlap. The T₃ axiom combines regularity with the T₀ condition (topological distinguishability of points), yielding a space where both point-from-point and point-from-closed-set separation are guaranteed. Some authors define T₃ as regular + T₁ instead; in either convention, the substantive content is the regularity condition itself.
Regularity sits one level above the Hausdorff condition in the separation hierarchy. The Hausdorff axiom (T₂) requires separating two points by disjoint open sets; regularity requires separating a single point from an entire closed set. This is a strictly stronger demand: a closed set can be infinite and its points can cluster near x, making the construction of disjoint open neighborhoods harder. Every regular T₁ space is Hausdorff — since singletons are closed in a T₁ space, regularity applied to x and the closed set {y} gives the T₂ separation. But T₂ does not imply regularity: there exist Hausdorff spaces where a point and a disjoint closed set cannot be separated by open sets, though such spaces are somewhat pathological.
Metric spaces are always regular, and the proof is constructive. If F is closed and x ∉ F, then d(x, F) = inf{d(x, y) : y ∈ F} > 0 (since F is closed and x is outside it). Setting r = d(x, F)/2, the open ball B(x, r) and the open set ∪{B(y, r) : y ∈ F} separate x from F: any point in both would be within r of x and within r of some point of F, giving d(x, y) < 2r = d(x, F), a contradiction. This argument illustrates a recurring theme: in metric spaces, the distance function provides explicit constructions for topological separation, which is why metric spaces satisfy all the standard separation axioms.
Regularity is preserved under several important constructions. Products of regular spaces are regular (in the product topology), and subspaces of regular spaces are regular. This makes regularity a robust property that persists as you build new spaces from existing ones. Regularity is the gateway to the stronger separation axiom of normality (T₄), which requires separating two disjoint closed sets from each other — a strictly harder task. The hierarchy T₂ ⊂ T₃ ⊂ T₄ represents progressively stronger geometric richness, with each level unlocking new theorems about the existence of continuous functions and extensions.