A compact set is a definite topological space with a finite subcover in its covering. In their topology, compact spaces in their properties can resemble a system of finite sets in the corresponding theory.
A compact set or compact is a subset of a topological space that is an induced type of compact space.
A relatively compact (precompact) set is only in the case of a compact closure. Given a convergent subsequence in space, it can be called sequentially compact.
A compact set has certain properties:
- a compact is an image of any continuous mapping;
- a closed subset always has compactness;
- a continuous one-to-one mapping, which is defined on a compact, refers to homeomorphism.
Examples of compact sets are:
- bounded and closed sets Rn;
- finite subsets in spaces that satisfy the axiom of divisibility T1;
- Ascoli-Arzela theorem characterizing a compact set for certain functional spaces;
- Stone space related to Boolean algebra;
- compactification of topological space.
Considering a universal set from the perspective of mathematics, it can be argued that this is a set that contains a collection of elements with specific properties. Along with the considered concept, there is also a hypothetical set that includes all kinds of components. However, its properties contradict the very essence of the set.
In the field of elementary arithmetic, a universal set is represented by a set of integers. However, a special role belongs to this set in set theory.
Many natural numbers include a set of elements (numbers) that can occur naturally during counting. There are two approaches to determining natural numbers:
- listing of items (first, second, etc.);
- the number of items (one, two, etc.).
Moreover, various non-integer and negative integers do not belong to the natural type of numbers. In the mathematical field, the set of natural numbers is denoted by N. This concept is infinite, due to the presence for any number of the natural type of a different natural number larger than the first.
Unlike natural numbers, integers are obtained as a result of performing mathematical operations on natural numbers such as addition or subtraction. The set of integers in mathematics is denoted by Z. According to the results of subtraction, addition and multiplication of two integer-type numbers, there will only be a number of the same type. The need for the appearance of this type of numbers is due to the lack of the ability to determine the difference between two natural numbers. It was Michael Shtifel who introduced negative numbers into mathematics.
It requires close attention to the consideration of such a concept as compact space. This term was introduced by P.S. Alexandrov to strengthen the concept of compact space, introduced into mathematics by M. Frechet. In the initial sense, a space of topological type is compact if there is a finite subcover in every open cover. With the subsequent development of mathematics, the term compactness became an order of magnitude higher than its lowest counterpart. And at present, it is bicompactness that is understood as compactness, and the old sense of the indicated term lies in the name “countably compact”. However, both concepts are equivalent when used in metric spaces.