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geodesic – injection
geodesic
A curve on a space is a geodesic if, given any two points on the curve, the segment of the curve joining the points is the shortest such curve. In Euclidean space, geodesics are straight lines. On a sphere, the geodesics are the great circles of the sphere.
 Graph Theory: A shortest path between two vertices is sometimes called a geodesic.
Cf. distance.
great circle
On a sphere, a great circle is a circle containing the endpoints of a diameter of the sphere, that is, a circle of maximal size. A geodesic on a sphere always lies on a great circle.
Hausdorff space
A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed.
Heine-Borel property
Every closed and bounded subset of the real numbers is compact.
Hempel’s Ravens Paradox
ARTICLE
A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition.
hexagon
A regular polygon having six equal sides and six equal interior angles. Hexagons can be used to tile the plane, and do so more efficiently than any other polygon.
Cf. tesselation.
homeomorphism
A homeomorphism is a bijective, continuous transformation of one topological space onto another whose inverse is also continuous.
homomorphism
A function f from one algebra to another is called a homomorphism if it preserves operations on elements, that is, if for any a, b in the domain, f(ab) = f(a)f(b).
Cf. group homomorphism, ring homomorphism.
hyperbola
The locus of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Like the ellipse and parabola, the hyperbola is a conic section.
Related article: Conics
hyperset
A set which is not well-founded, i.e., which involves self-membership or, equivalently, an infinite descending membership chain. Example: the Quine atom x = {x}.
hypotenuse
On a right triangle, the side opposite the right angle.
icosahedron
A polyhedron having twenty faces.
The faces of a regular icosahedron are congruent, equilateral triangles.
Cf. Platonic solid.
identity function
A function that maps each domain element to itself. Also called the identity map.
identity map
See identity function.
image
Given a function f with domain X, the image under f of a subset A of X, denoted f(A), is the subset of the range consisting of those elements to which elements of A are mapped by f.
inaccessible cardinal
A cardinal k is called weakly inaccessible if and only if it is a regular limit cardinal. A cardinal k is called strongly inaccessible if and only if k > w and for every cardinal l less than k, 2l is less than k. If GCH is assumed, then weakly inaccessibles are strongly inaccessible. Ordinary set theory (ZFC) cannot prove that weakly inaccessibles exist.
Cf. König’s Lemma.
independent axiom
In a formal mathematical theory, a formula or statement in the theory is said to be independent if it cannot be derived (proved, deduced) from the other axioms of the theory.
inf
Abbreviation of infimum.
infimum
The infimum of any subset of a linear order (linearly ordered set) is the greatest lower bound of the subset. In particular, the infimum of any set of numbers is the largest number in the set which is less than or equal to every other number in the set. In a complete linear order the infimum of any bounded set always exists.
Cf. supremum, least upper bound axiom.
infinity
Infinity is a concept understood in different ways depending upon the context in which the word is used. In particular, infinity is not a number in the ordinary sense. The so-called extended real numbers include either a positive or a negative infinity (but not both). When this is done, the algebraic forms "infinity plus negative infinity", "infinity times zero", and "infinity divided by infinity" are undefined. Infinite ordinals may be countable or uncountable. Whether actually infinite totalities may be admitted, used, or analyzed remains a contentious issue in the philosophy of mathematics. See the minitext for a thorough treatment.
Related MiniText: Infinity -- You Can't Get There From Here...
injection
An injective function, i.e., a function that is “one-to-one.” Equivalently, a function that maps exactly one element of its domain to each element of its range.
Cf. surjection, bijection.
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