compact set is a certain topological space in the coating which is a finite sub-covering.Compact spaces in the topology of their properties may resemble a system of finite sets in the corresponding theory.
compact set or CD - a subset of a topological space, which is induced by the type of compact space.
relatively compact (precompact) set is only in the case of a compact circuit.When you select a subsequence converging in the space, it can be called sequentially compact.
compact set has certain properties:
- is a compact way any continuous mapping;
- closed subset always has a compact;
- a continuous one-one mapping that is defined on the compact relates to homeomorphism.
examples of compact sets are:
- bounded and closed sets Rn;
- finite subsets in spaces that satisfy the axiom of divisibility T1;
- Ascoli theorem Arzela characterizing compact set for certain function spaces;
- Stone space belonging to the Boolean algebra;
- compactification of a topological space.
Considering the universal set to the position of mathematics, it can be argued that this set that contains a set of elements with specific properties.In addition to considering the concept there is a hypothetical set includes various components.However, its properties are contrary to the very essence of the set.
In the field of elementary arithmetic universal set is represented by a set of integers.However, a special role belongs to this set in set theory.
set of natural numbers contains a set of elements (numbers) that can occur naturally during counting.There are two approaches in determining the natural numbers:
- listed items (first, second, etc.);
- number of subjects (one, two, etc.).
This is not different integers and negative integers to the natural type of numbers do not apply.In the mathematical field of set of natural numbers is N. This notion is endless, thanks to the presence of any number of different types of natural natural number greater than the first.
Unlike natural, integers are the result of the implementation of such mathematical operations on the natural numbers as addition or subtraction.The set of integers in mathematics is designated Z. By the results of the subtraction, addition and multiplication of two numbers is the number of a type only of the same type.The need for the appearance of this type of numbers due to the lack of ability to identify the difference of two positive integers.That Michael Stiefel introduced negative numbers in mathematics.
Requires attention considering such thing as a compact space.This term was introduced by PSAlexandrov to reinforce the notion of a compact space, introduced in mathematics M. Fréchet.In the original sense of the topological type of a compact space in the event of a final subcovering each open cover.In the subsequent development of mathematics, the term compactness became an order of magnitude higher than its lower counterpart.And now it is understood by the compactness compactness, and the old sense of the term is in the title of "countably compact."However, both concepts are equivalent when used in metric spaces.
