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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 ...

If f:(X,A)->(Y,B) is homotopic to g:(X,A)->(Y,B), then f_*:H_n(X,A)->H_n(Y,B) and g_*:H_n(X,A)->H_n(Y,B) are said to be the induced maps.

A geometry constructed without reference to measurement. The only primitive concepts are those of points and intermediacy. There are 10 axioms underlying ordered geometry.

The law appearing in the definition of Boolean algebras and lattice which states that a ^ (a v b)=a v (a ^ b)=a for binary operators v and ^ (which most commonly are logical ...

A bilinear form on a real vector space is a function b:V×V->R that satisfies the following axioms for any scalar alpha and any choice of vectors v,w,v_1,v_2,w_1, and w_2. 1. ...

There are at least two distinct notions of linear space throughout mathematics. The term linear space is most commonly used within functional analysis as a synonym of the ...

A topological space fulfilling the T_2-axiom: i.e., any two points have disjoint neighborhoods. In the terminology of Alexandroff and Hopf (1972), a T_2-space is called a ...

Let union represent "or", intersection represent "and", and ^' represent "not." Then, for two logical units E and F, (E union F)^'=E^' intersection F^' (E intersection ...

The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers aleph_0 and the "large" ...

Propositional calculus is the formal basis of logic dealing with the notion and usage of words such as "NOT," "OR," "AND," and "implies." Many systems of propositional ...