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One of the Eilenberg-Steenrod axioms. It states that, for every pair (X,A), there is a natural long exact sequence ...->H_n(A)->H_n(X)->H_n(X,A)->H_(n-1)(A)->..., where the ...

One of the Eilenberg-Steenrod axioms which states that, if f:(X,A)->(Y,B) is homotopic to g:(X,A)->(Y,B), then their induced maps f_*:H_n(X,A)->H_n(Y,B) and ...

The theory of natural numbers defined by the five Peano's axioms. Paris and Harrington (1977) gave the first "natural" example of a statement which is true for the integers ...

Axiomatic set theory is a version of set theory in which axioms are taken as uninterpreted rather than as formalizations of pre-existing truths.

A logical system which possesses an explicitly stated set of axioms from which theorems can be derived.

Parallel lines are everywhere equidistant. This postulate is equivalent to the parallel postulate.

Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter ...

An experiment E(S,F,P) is defined (Papoulis 1984, p. 30) as a mathematical object consisting of the following elements. 1. A set S (the probability space) of elements. 2. A ...

One of the Eilenberg-Steenrod axioms. Let X be a single point space. H_n(X)=0 unless n=0, in which case H_0(X)=G where G are some groups. The H_0 are called the coefficients ...

Presburger arithmetic is the first-order theory of the natural numbers containing addition but no multiplication. It is therefore not as powerful as Peano arithmetic. ...