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A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: 1. The empty set ...

Definitions about a set which depend on the entire set.

A group G is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of ...

The set R union {infty}, obtained by adjoining one improper element to the set R of real numbers, is the set of projectively extended real numbers. Although notation is not ...

Given any open set U in R^n with compact closure K=U^_, there exist smooth functions which are identically one on U and vanish arbitrarily close to U. One way to express this ...

A set considered together with the sigma-algebra on the set.

Let F be the set of complex analytic functions f defined on an open region containing the closure of the unit disk D={z:|z|<1} satisfying f(0)=0 and df/dz(0)=1. For each f in ...

An operator defined on a set S which takes two elements from S as inputs and returns a single element of S. Binary operators are called compositions by Rosenfeld (1968). Sets ...

If V and W are Banach spaces and T:V->W is a bounded linear operator, the T is said to be a compact operator if it maps the unit ball of V into a relatively compact subset of ...

A extension ring (or ring extension) of a ring R is any ring S of which R is a subring. For example, the field of rational numbers Q and the ring of Gaussian integers Z[i] ...