Set closures. Closed and open sets

Let us now prove some special properties of closed and open sets.

Theorem 1. The sum of a finite or countable number of open sets is an open set. The product of a finite number of open sets is an open set,

Consider the sum of a finite or countable number of open sets:

If , then P belongs to at least one of Let Since is an open set, then some -neighborhood of P also belongs to The same -neighborhood of P also belongs to the sum g, whence it follows that g is an open set. Consider now the final product

and let P belong to g. Let us prove, as above, that some -neighborhood P belongs to g. Since P belongs to g, then P belongs to all . Since are open sets, then for any there is some -neighborhood of the point belonging to . If the number is taken equal to the smallest of the number of which is finite, then the -neighborhood of the point P will belong to all and, consequently, to g. Note that one cannot assert that the product of a countable number of open sets is an open set.

Theorem 2. The set CF is open and the set CO is closed.

Let us prove the first assertion. Let P belong to CF. It is necessary to prove that some neighborhood P belongs to CF. This follows from the fact that if there were points F in any -neighborhood P, the point P, which does not belong by condition, would be the limit point for F and, due to its closedness, would have to belong, which leads to a contradiction.

Theorem 3. The product of a finite or countable number of closed sets is a closed set. The sum of a finite number of closed sets is a closed set.

Let us prove, for example, that the set

closed. Passing to additional sets, we can write

By the theorem, open sets, and, by Theorem 1, the set is also open, and thus the complementary set g is closed. Note that the sum of a countable number of closed sets may also be a non-closed set.

Theorem 4. A set is an open set and a closed set.

It is easy to check the following equalities:

From them, by virtue of the previous theorems, Theorem 4 follows.

We will say that a set g is covered by a system M of some sets if every point g is included in at least one of the sets of the system M.

Theorem 5 (Borel). If a closed bounded set F is covered by an infinite system a of open sets O, then from this infinite system one can extract a finite number of open sets that also cover F.

We prove this theorem from the converse. Let us assume that no finite number of open sets from the system a covers and reduce this to a contradiction. Since F is a bounded set, then all points of F belong to some finite two-dimensional interval. Let us divide this closed interval into four equal parts, dividing the intervals in half. Each of the obtained four intervals will be taken closed. Those points of F that fall on one of these four closed intervals will, by virtue of Theorem 2, represent a closed set, and at least one of these closed sets cannot be covered by a finite number of open sets from the system a. We take one of the above four closed intervals where this circumstance takes place. We again divide this interval into four equal parts and argue in the same way as above. Thus, we obtain a system of nested intervals, of which each next one is the fourth part of the previous one, and the following circumstance takes place: the set of points F belonging to any k cannot be covered by a finite number of open sets from the system a. With an infinite increase in k, the gaps will shrink indefinitely to some point P, which belongs to all the gaps. Since for any k they contain an uncountable set of points, the point P is a limit point for and therefore belongs to F, since F is a closed set. Thus the point P is covered by some open set belonging to the system a. Some -neighborhood of the point P will also belong to the open set O. For sufficiently large values ​​of k, the intervals D will fall inside the above -neighborhood of the point P. Thus, these will be completely covered by only one open set O of the system a, and this contradicts the fact that the points belonging to for any k cannot be covered by a finite number of open sets belonging to a. Thus the theorem is proved.

Theorem 6. An open set can be represented as the sum of a countable number of half-open gaps in pairs without common points.

Recall that a half-open gap in the plane is a finite gap defined by inequalities of the form .

Let's put on the plane a grid of squares with sides parallel to the axes, and with a side length equal to one. The set of these squares is a countable set. We choose from these squares those squares all of whose points belong to a given open set O. The number of such squares may be finite or countable, or there may be no such squares at all. We divide each of the remaining squares of the grid into four identical squares and from the newly obtained squares we again select those whose all points belong to O. We again divide each of the remaining squares into four equal parts and select those squares whose all points belong to O, etc. Let us show that any point P of the set O will fall into one of the chosen squares, all of whose points belong to O. Indeed, let d be a positive distance from P to the boundary of O. When we reach squares whose diagonal is less than , then we can obviously assert that the point P has already fallen into a square, all of whose volumes belong to O. If the chosen squares are considered half-open, then they will not have pairwise common points, and the theorem is proved. The number of selected squares will necessarily be countable, since the finite sum of half-open gaps is obviously not an open set. Denoting by DL those half-open squares that we obtained as a result of the above construction, we can write

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

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

A countable set is the "smallest" infinite set, that is, any infinite set has 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, Integer numbers, Rational numbers, Algebraic numbers, Ring of periods, Computable numbers, Arithmetic numbers.

Theory of real numbers.

(Real = real - a reminder for us guys.)

The set R contains rational and irrational numbers.

Real numbers that are not rational are called irrational.

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 the two relations takes place a or a>b