Compact set

A compact set is a definite topological space in whose covering there is a finite subcover. Compact spaces in topology in their properties can resemble a system of finite sets in the corresponding theory.

A compact set or a compact subset of a topological space that is an induced type of a compact space.

A relatively compact (precompact) set is only in the case of a compact closure. When a convergent subsequence is singled out in a space, it can be called sequentially compact.

A compact set has certain properties:

- a compactum is the image of any continuous mapping;

A closed subset always has compactness;

- continuous one-to-one mapping, which is defined on a compactum, refers to a homeomorphism.

Examples of a compact set are:

- bounded and closed sets Rn;

- finite subsets in spaces that satisfy the axiom of divisibility of T1;

- Ascoli-Arzela theorem characterizing a compact set for certain function spaces;

- Stone space related to Boolean algebra;

Compactification of a topological space.

Considering the universal set from the position of mathematics, it can be argued that this set, which contains a set of elements with specific properties. Along with the concept considered, there is also a hypothetical set including all possible components. However, its properties contradict the very essence of the set.

In the sphere of elementary arithmetic, the universal set is represented by a collection of integers. However, a special role belongs to this set in set theory.

The set of natural numbers includes a set of elements (numbers) that can arise naturally during counting. There are two approaches to determining natural numbers:

- transfer of items (first, second, etc.);

- the number of items (one, two, etc.).

In this case, different non-integer and negative integers to the natural type of numbers do not apply. In the mathematical sphere, the set of natural numbers is denoted by N. This concept is infinite because of the presence for any number of natural type of another natural number greater than the first.

Unlike natural numbers, integers are obtained by performing such mathematical operations on natural numbers as addition or subtraction. The set of integers in mathematics is denoted by Z. By the results of subtraction, addition and multiplication of two integers of integer type there will be a number of only the same type. The necessity of the appearance of this type of numbers is due to the lack of the ability to determine the difference of two natural numbers. It was Michael Stiefel who introduced negative numbers into mathematics.

It requires close attention to the consideration of such a notion as a compact space. This term was introduced by P.S. Aleksandrov for strengthening the concept of a compact space introduced in the mathematics of M. Frechet. In the original understanding, a space of topological type is compact in the case of a finite subcovering in each open cover. With the subsequent development of mathematics, the term bicompactness became an order of magnitude higher than its lower analog. And at the present time it is bicompactness that is understood as compactness, and the old meaning of the term is "countably compact". However, both concepts are equivalent when used in metric spaces.