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A property of a space which is also true of each of its subspaces. Being "first-countable" is hereditary, but having a given genus is not.

An axiomatic system is said to be categorical if there is only one essentially distinct representation for it. In particular, the names and types of objects within the system ...

Let A and B_j be sets. Conditional probability requires that P(A intersection B_j)=P(A)P(B_j|A), (1) where intersection denotes intersection ("and"), and also that P(A ...

A map is called bijective if it is both injective and surjective. A bijective map is also called a bijection. A function f admits an inverse f^(-1) (i.e., "f is invertible") ...

Every Boolean algebra is isomorphic to the Boolean algebra of sets. The theorem is equivalent to the maximal ideal theorem, which can be proved without using the axiom of ...

A product of ANDs, denoted ^ _(k=1)^nA_k. The conjunctions of a Boolean algebra A of subsets of cardinality p are the 2^p functions A_lambda= union _(i in lambda)A_i, where ...

Given a sequence of real numbers a_n, the infimum limit (also called the limit inferior or lower limit), written lim inf and pronounced 'lim-inf,' is the limit of ...

Model completion is a term employed when existential closure is successful. The formation of the complex numbers, and the move from affine to projective geometry, are ...

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups. The matrix direct sum is defined by ...

Pre-Calculus