regular,
adj. 1a. (of a geometric figure) having all sides and all angles equal, as in a regular polygon such as the octagon in the figure below.
b. (of a geometric solid) having regular polygons as bases. A regular polyhedron, such as the regular tetrahedron shown in perspective in the second figure below, has identical regular polygons as bases, making equal angles with one another: a regular prism has regular polygons as bases; and a regular pyramid has a regular polygon as its base and its vertex perpendicularly above the center of the base.
Regular octagon and regular tetrahedron.
2. (of a complex function) another word for analytic. 3. (of a topological space) such that a point and a disjoint closed set can be separated by open sets; equivalently, for every neighborhood of a point there is another neighborhood of the point whose closure is contained in the given neighborhood. Complete regularity is characterized by separation by continuous functionals: if p ∉V with V closed in X, there is a continuous function f : X → [0, 1] such that f(V) = 1 and f(p) = 0. Compare normal, T-axioms, Tietze's extension theorem. 4. (of a curve) possessing no singular point, having only ordinary points. 5. (of a summation method) giving the correct sum to a convergent series or sequence, as opposed to conservative methods that maintain convergence but that may change the value of the limit. See Tauberian condition. See also Abel summation, Cesaro summation. 6. (of an outer measure) such that every set, E, is contained in a measurable subset, A, of the same measure: μ(A) = μ*(E). 7. (of a Borel measure on a locally compact Hausdorff space) assigning a finite measure to every compact set and such that
(i) the measure of any Borel set is the infimum of the measures of open measurable sets containing the given set, and
(ii) the measure of every open set is the supremum of the measures of compact sets contained in the given set.
When X is compact, (i) implies (ii), and if each open subset of X is sigma-compact (as when X is compact metrizable), it suffices for each compact set to have finite measure. 8. (of a graph) having every vertex of the same degree. 9. (of an action of a group on a set) such that the set has precisely one orbit under the group action, and the stabilizer of every element of the set is trivial.
10. (of an element of a ring) such that its left or right product with any non-zero element of the ring is non-zero; that is, x is regular if and only if, for all r ∈R, either rx = 0 or xr = 0 only if r = 0. For example, every non-zero element of an integral domain is regular.
b. (of a geometric solid) having regular polygons as bases. A regular polyhedron, such as the regular tetrahedron shown in perspective in the second figure below, has identical regular polygons as bases, making equal angles with one another: a regular prism has regular polygons as bases; and a regular pyramid has a regular polygon as its base and its vertex perpendicularly above the center of the base.
2. (of a complex function) another word for analytic. 3. (of a topological space) such that a point and a disjoint closed set can be separated by open sets; equivalently, for every neighborhood of a point there is another neighborhood of the point whose closure is contained in the given neighborhood. Complete regularity is characterized by separation by continuous functionals: if p ∉V with V closed in X, there is a continuous function f : X → [0, 1] such that f(V) = 1 and f(p) = 0. Compare normal, T-axioms, Tietze's extension theorem. 4. (of a curve) possessing no singular point, having only ordinary points. 5. (of a summation method) giving the correct sum to a convergent series or sequence, as opposed to conservative methods that maintain convergence but that may change the value of the limit. See Tauberian condition. See also Abel summation, Cesaro summation. 6. (of an outer measure) such that every set, E, is contained in a measurable subset, A, of the same measure: μ(A) = μ*(E). 7. (of a Borel measure on a locally compact Hausdorff space) assigning a finite measure to every compact set and such that
(i) the measure of any Borel set is the infimum of the measures of open measurable sets containing the given set, and
(ii) the measure of every open set is the supremum of the measures of compact sets contained in the given set.
When X is compact, (i) implies (ii), and if each open subset of X is sigma-compact (as when X is compact metrizable), it suffices for each compact set to have finite measure. 8. (of a graph) having every vertex of the same degree. 9. (of an action of a group on a set) such that the set has precisely one orbit under the group action, and the stabilizer of every element of the set is trivial.
10. (of an element of a ring) such that its left or right product with any non-zero element of the ring is non-zero; that is, x is regular if and only if, for all r ∈R, either rx = 0 or xr = 0 only if r = 0. For example, every non-zero element of an integral domain is regular.