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A topological space X has a one-point compactification if and only if it is locally compact. To see a part of this, assume Y is compact, y in Y, X=Y\{y} and x in X. Let C be ...

A right or left ideal of a ring. The term is used especially in noncommutative rings to denote a right ideal that is not a left ideal, or conversely.

A map is called "one-to-one" (or "1:1") if it is injective. However, being in "one-to-one correspondence" is synonymous with being a bijection. Unfortunately, this confusing ...

"A and B are in one-to-one correspondence" is synonymous with "A and B are bijective."

A collection of open sets of a topological space whose union contains a given subset. For example, an open cover of the real line, with respect to the Euclidean topology, is ...

An open interval is an interval that does not include its end points. The open interval {x:a<x<b} is denoted (a,b), although the nonstandard notation ]a,b[ is sometimes also ...

A noncompact manifold without boundary.

A map which sends open sets to open sets.

In a topological space X, an open neighborhood of a point x is an open set containing x. A set containing an open neighborhood is simply called a neighborhood.

Given an acute angle in a right triangle, the opposite side is the leg of the triangle which is not incident on the angle. Lengths of opposite and adjacent sides appear ...