Open and closed sets. Closures of sets. Closed and open sets Closedness of actions on the set of natural numbers

CLOSED SET

in the topological space - containing all its limit points. Thus, all points of the complement of the 3. m. are internal, and therefore the 3. m. can be defined as open. The concept of 3.m. underlies the definition of topological. space as a non-empty set X with a given system of sets (called closed) satisfying the axioms: all X and are closed; any number 3. m. is closed; finite number 3. m is closed.

Lit: Kuratovsky K., Topology, [trans. from English], vol. 1, M., 1966.

A. A. Maltsev.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what a “CLOSED SET” is in other dictionaries:

closed set- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information technology in general EN closed set ... Technical Translator's Guide

For the term "Closedness" see other meanings. A closed set is a subset of a space whose complement is open. Contents 1 Definition 2 Closure 3 Properties ... Wikipedia

A set that is open (closed) with respect to a certain set E, a set Mtopological. space X such that (the overbar means the closure operation). In order for a certain set to be open (closed) with respect to E, it is necessary and... ... Mathematical Encyclopedia

Subset of topological a space that is both open and enclosed. Topological a space X is disconnected if and only if it contains a space different from X and from O.Z. m. If the family of all O. z. m. topological space is... ... Mathematical Encyclopedia

Or the catlocus of a point in a Riemannian manifold is a subset of points through which no shortest path passes. Contents 1 Examples ... Wikipedia

For the mathematical concept of the same name, see Closed set and Space (mathematics) Storm sewer ... Wikipedia

Books

• Limit theorems for associated random fields and related systems, Alexander Bulinsky. The monograph is devoted to the study of the asymptotic properties of a wide class of stochastic models arising in mathematical statistics, percolation theory, statistical physics and theory...

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I know mathematical analysis In the first year of university there are many incomprehensible and unusual things. One of the first of these “new” topics is open and closed sets. We will try to provide explanations on this topic.

Before proceeding with the formulation of definitions and problems, let us recall the meaning of the notation used and quantifiers :
∈ - belongs
∅ — empty set
Ε - set of real numbers
x* - fixed point
A* - set of boundary points
: - such that
⇒ — therefore
∀ - for each
∃ - exists
U ε (x) — neighborhood of x by ε
Uº ε (x) - punctured neighborhood of x with respect to ε

So,
Definition 1: A set M ∈ Ε is called open if for any y ∈ M there is an ε > 0 such that the neighborhood of y in ε is strictly less than M
Using quantifiers, the definition will be written as follows:
M ∈ Ε is open if ∀ y∈M ∃ ε>0: U ε (y)< M

In simple terms, an open set consists of interior points. Examples of an open set are the empty set, line, interval (a, b)

Definition 2: A point x* ∈ E is called a boundary point of a set M if any neighborhood of the point x contains points from both the set M and its complement.
Now using quantifiers:
x*∈ E is a boundary point if ∀U ε (x) ∩ M ≠ ∅ and ∀U ε (x) ∩ E\M

Definition 3: A set is called closed if it contains all boundary points. Example - segment

It is worth noting that there are sets that are both open and closed. This is, for example, the entire set of real numbers and the empty set (later it will be proven that these are 2 possible and only cases).

Let us prove several theorems related to open and closed sets.

Theorem 1: Let the set A be open. Then the complement to the set A is a closed set.

B = E\A

Let us assume that B is not closed. Then there is a boundary point x* that does not belong to B, and therefore belongs to A. By the definition of a boundary point, the neighborhood of x* has an intersection with both B and A. However, on the other hand, x* is an interior point of the open set A, therefore the entire neighborhood of the point x* lies in A. From here we conclude that the sets A and B do not intersect in an empty set. This cannot be, therefore our assumption is incorrect and B is a closed set, etc.
In quantifiers, the proof can be written more briefly:
Suppose that B is not closed, then:
(1) ∃ x∈A*:x∈A ⇒ ∀U ε (x) ∩ B ≠ ∅ (definition of boundary point)
(2) ∃ x∈A*:x∈A ⇒ ∀U ε (x) ⊂ A ≠ ∅ (definition of open set)
From (1) and (2) ⇒ A ∩ B ≠ ∅. But A ∩ B = A ∩ E\A = 0. A contradiction. B - closed, etc.

Theorem 2: Let the set A be closed. Then the complement to the set A is an open set.
Proof: Let us denote the complement of set A as set B:
B = E\A
We will prove it by contradiction.
Let us assume that B is a closed set. Then any boundary point lies in B. But since A is also a closed set, all boundary points belong to it. However, a point cannot simultaneously belong to a set and its complement. Contradiction. B is an open set, etc.
In quantifiers it will look like this:
Let us assume that B is closed, then:
(1) ∀ x∈A*:x∈A (from the condition)
(1) ∀ x∈A*:x∈B (from assumption)
From (1) and (2) ⇒ A ∩ B ≠ ∅. But A ∩ B = A ∩ E\A = 0. A contradiction. B - open, etc.

Theorem 3: Let the set A be closed and open. Then A = E or A = ∅
Proof: Let’s start writing it down in detail, but I’ll immediately use quantifiers.
Suppose that the set C is closed and open, with C ≠ ∅ and C ≠ E. Then it is obvious that C ⊆ E.
(1) ∃ x∈A*:x∈C ⇒ ∀U ε (x) ∩ E\C ≠ ∅ (definition of the boundary point that belongs to C)
(2) ∃ x∈A*:x∈A ⇒ ∀U ε (x) ⊂ B (definition of an open set C)
From (1) and (2) it follows that E\C ∩ C ≠ ∅, but this is false. Contradiction. C cannot be both open and closed at the same time, etc.

Mathematical analysis is fundamental mathematics, complex and unusual for us. But I hope something became clearer after reading the article. Good luck!

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A countable set is an infinite set whose elements can be numbered by natural numbers, or it is a set equivalent to the set of natural numbers.

Sometimes sets of equal cardinality to any subset of the set of natural numbers are called countable, that is, all finite sets are also considered countable.

A countable set is the “smallest” infinite set, that is, in any infinite set there is a countable subset.

Properties:

1. Any subset of a countable set is at most countable.

2. The union of a finite or countable number of countable sets is countable.

3. The direct product of a finite number of countable sets is countable.

4. The set of all finite subsets of a countable set is countable.

5. The set of all subsets of a countable set is continuous and, in particular, is not countable.

Examples of countable sets:

Prime numbers Natural numbers, Integers, Rational numbers, Algebraic numbers, Period ring, Computable numbers, Arithmetic numbers.

Theory of real numbers.

(Real = real - reminder for us guys.)

The set R contains rational and irrational numbers.

Real numbers that are not rational are called irrational numbers

Theorem: There is no rational number whose square is equal to the number 2

Rational numbers: ½, 1/3, 0.5, 0.333.

Irrational numbers: root of 2=1.4142356…, π=3.1415926…

The set R of real numbers has the following properties:

1. It is ordered: for any two different numbers a and b one of two relations holds a or a>b