Theorem 3.1. The union of any number of open sets is an open set.
Let G k, where k О N are open sets.
3Choose any point X O OG. By the definition of a union of sets, a point X o belongs to one of the sets G k. Insofar as G k is an open set, then there exists e- point neighborhood x o, which lies entirely in the set Gk: U(x o , e)Ì G k Þ U(x o ,e)ÌG.
Got that any point x o ÎG internal, which means that G is an open set. 4
Theorem 3.2 . The intersection of a finite number of open non-empty sets is an open set.
Let G k (k = 1,2, …,n) are open sets.
Let us prove that is an open set.
3Choose any point X O OG. By definition of intersection of sets X o belongs to each of the sets G k. Because every set G k open, then in any set G k exists e k- neighborhood of the point X O : U(x o , e k)М G k. set of numbers ( e 1 , e 2 ,…, e n) is finite, so there is a number e = min{e 1 ,e 2 ,…,e n). Then e- neighborhood of the point X o is in every e k- neighborhood of the point X O :U(x o , e)ÌU e(x o , e k) U U(x o , e)ÌG.
Got that X o is an interior point of the set G, which means that G is an open set. 4
Remark 3.1. The intersection of an infinite set of open sets may not be an open set.
Example 3.1. Let in space R G k =(2 – 1/k; 4+ 1/k), where k= 1,2,…,n,…. G 1 =(1;5), G2(1.5;4.5), Segment М G k and is not an open set, points 2 and 4 are not interior.
Theorem 3.3 . The intersection of any collection of closed non-empty sets is a closed set.
Let F k are closed sets.
Let us prove that the set is closed, i.e. it contains all its limit points.
3Let X F. It follows from the definition of the intersection of sets that in any e- neighborhood of the point X o there are infinitely many points of each of the sets F k, which means that X o is the limit point of each set F k. Due to the closedness of the sets F k dot
X O Î F k "k Þ x O О F. Since the point X F, which means many F closed. 4
Theorem 3.4. The union of a finite number of closed sets is a closed set.
Let each set F k closed.
Let us prove that the set is closed, i.e., if X o is the limit point of the set F, then X O О F.
3Let X o - any limit point of the set F, then in any e- neighborhood of the point X o there are infinitely many points of the set . Since the number of sets Fk final, then X o belongs to at least one of the sets Fk, i.e. X o is the limit point for this set.
Due to the closed F k dot X oh belongs Fk, and therefore to the set . Since the point X o is chosen arbitrarily, then all limit points belong to the set F, which means many F closed. 4
Remark 3.2. The union of an infinite number of closed sets can be an open set.
Example 3.2 . In space R: F k =
F 1 =; F2 = ; …. The interval (2;5) is an open set.
We accept without proof Theorems 3.5 and 3.6 related to the complement of the set E to many X: C x E=CE.
Theorem 3.5 . If the set E closed, then its complement CE open set.
Example 3.3 . E=, C R E =(- ¥, 2)È (5,+¥ ).
Theorem 3.6 . If the set E open, then its complement CE closed set.
Example 3.4 . E=(2,5), C R E =(-¥, 2]È[ 5, +¥ ).
Functions of several variables.
When studying many phenomena, one has to meet with functions of two or more independent variables.
Examples.
1) Area of a rectangle with sides x and y: S=xy.
2) Volume of a cuboid with edges x,y,z: V=xyz.
3) According to Ohm's law, the voltage U in the electric circuit is related to the resistance R of the circuit and the current strength I by the dependence U=RI. If we consider U and R as data, then I is defined as a function of U and R: I= .
The elements of the arithmetic space R n are ordered sets of n real numbers(x 1, x 2,…, x n). These ordered sets are called points in an n-dimensional space or n-dimensional vectors.
x \u003d (x 1, x 2, ..., x n), y \u003d (y 1, y 2, ..., y n). x 1, x 2,…, x n – point coordinates.
Definition. The distance between the points x \u003d (x 1, x 2, ..., x n) and y \u003d (y 1, y 2, ..., y n):
d(x,y)= 
(1)
Distance Properties:
1) d(x,y)³0, moreover, d(x,y)=0 Û x=y, i.e. x i \u003d y i "i \u003d 1,2, ..., n.
2) d(x,y)=d(y,x) is a symmetry property.
3) d(x,y)£d(x,z)+d(z,y) "x,y,zнR n is the triangle inequality ( £ + ).
Let a(а 1 ,а 2 ,…,а n) be an arbitrary point in the space R n and let R>0 be some number. The set of all points x(x 1, x 2,…, x n):
В(a,R)=(xОR n: d(x,a) (a,R)=(xОR n: d(x,a)£R) is a closed ball (sphere) with center at a and radius R. S(a,R)=(xнR n: d(x,a)=R) is a sphere in R n . Therefore, the equation of the sphere in R n is: Definition. Let there be numbers a 1 ,…,a n and b 1 ,…,b n such that a 1
called open parallelepiped - P. The set of all points M(х 1 ,х 2 ,…,х n)нR n for which called closed parallelepiped -. Point С( ,…, ) – center of the parallelepiped. An open sphere of any radius R>0 centered at the point M 0 ( ,…, ) can be considered as neighborhood this point. (Similarly, an open parallelepiped centered at the point M 0 ( ,…, ) can be considered as a neighborhood). Definition. Let Е be some set of points from R n . The set E is called limited, if there exists a number R>0 such that all points of the set E are inside a sphere of radius R centered at the point O(0,…,0). Theorem. Let the set E(M)ÌR n . Let (x 1 ) - set, which is formed by the first coordinates of the points МОЕ, ………………………………………………………………………….. (x n ) - the set that is formed by the n-th coordinates of the points МОЕ. In order for the set E(M) to be bounded, it is necessary and sufficient that the sets (x 1 ),..., (x n ) be bounded at the same time. Proof. Need. Let E(M) be bounded. Therefore, there exists a number R>0 such that d(M,O) 0£êx 1 ê£ And this means that the sets (x 1 ),..., (x n ) are bounded. Adequacy. Let the sets (x 1 ),..., (x n ) be bounded. Hence $C>0: êx 1 ê i.e., d(M,O) Definition. The set is called open, if each point of this set is included in it together with its neighborhood. Properties of open sets. 1) the sets R n and Æ are open. 2) The union of any system of open sets is open (show). 3) The intersection of a finite system of open sets is open (show). The point M 0 ОЕ is called condensation point of the set EÌR n if each of its neighborhoods contains at least one point of the set E that is different from M 0 . Definition. The set FÌR n is called closed, if its complement in R n is open (that is, if R n \F is open). Condensation points of an open set that do not belong to it are called border points this set. The boundary points form border sets. An open set with its own boundary is called closed. One of the main tasks of the theory of point sets is the study of the properties of various types of point sets. Let's get acquainted with this theory on two examples and study the properties of the so-called closed and open sets. The set is called closed
if it contains all of its limit points. If a set has no limit points, then it is also considered to be closed. In addition to its limit points, a closed set can also contain isolated points. The set is called open
if each of its points is internal to it. Let's bring examples of closed and open sets
. Every segment is a closed set, and every interval (a, b) is an open set. Improper half-intervals and closed, and improper intervals and open. The entire line is both a closed and an open set. It is convenient to think of the empty set as both closed and open at the same time. Any finite set of points on a line is closed, since it has no limit points. A set consisting of points: closed; this set has a single limit point x=0, which belongs to the set. The main task is to find out how an arbitrary closed or open set works. To do this, we need a number of auxiliary facts, which we will accept without proof. Let E be an arbitrary set of points on the line. We call the complement of the set E and denote by CE the set of all points on the line that do not belong to the set E. It is clear that if x is an external point for E, then it is an internal point for the set CE and vice versa. 4. If the set F is closed, then its complement CF is open and vice versa. Proposition 4 shows that there is a very close connection between closed and open sets: one is the complement of the other. Because of this, it suffices to study only one closed or one open set. Knowing the properties of sets of one type allows you to immediately find out the properties of sets of another type. For example, any open set is obtained by removing some closed set from a line. We proceed to the study of the properties of closed sets. We introduce one definition. Let F be a closed set. An interval (a, b) with the property that none of its points belongs to the set F, while the points a and b belong to F, is called an adjacent interval of the set F. Among adjacent intervals, we will also include improper intervals or, if the point a or the point b belongs to the set F, and the intervals themselves do not intersect with F. Let us show that if a point x does not belong to a closed set F, then it belongs to one of its adjacent intervals. Denote by the part of the set F located to the right of the point x. Since the point x itself does not belong to the set F, it can be represented in the form of an intersection: Each of the sets F and is closed. Therefore, by Proposition 1, the set is closed. If the set is empty, then the entire half-interval does not belong to the set F. Let us now assume that the set is not empty. Since this set lies entirely on the half-interval, it is bounded from below. Denote by b its lower bound. According to Proposition 3, which means Further, since b is the infimum of the set, then the half-interval (x, b) lying to the left of the point b does not contain points of the set and, therefore, does not contain points of the set F. Thus, we have constructed a half-interval (x, b) that does not contain points of the set F, and either, or the point b belongs to the set F. Similarly, a half-interval (a, x) is constructed that does not contain points of the set F, and either, or. Now it is clear that the interval (a, b) contains the point x and is an adjacent interval of the set F. It is easy to see that if and are two adjacent intervals of the set F, then these intervals either coincide or do not intersect. It follows from the above that any closed set on the line is obtained by removing from the line a certain number of intervals, namely, adjacent intervals of the set F. Since each interval contains at least one rational point, and all rational points on the line are a countable set, it is easy make sure that the number of all adjacent intervals is at most countable. From here we get the final conclusion. Any closed set on a line is obtained by removing from the line at most a countable set of disjoint intervals. By Proposition 4, this immediately implies that any open set on the line is at most a countable sum of disjoint intervals. By virtue of Propositions 1 and 2, it is also clear that any set arranged as indicated above is indeed closed (open). As can be seen from the following example, closed sets can have a very complex structure. One of the main tasks of the theory of point sets is the study of the properties of various types of point sets. We will introduce the reader to this theory with two examples. Namely, we will study here the properties of the so-called closed and open sets. A set is called closed if it contains all of its limit points. If a set does not have a single limit point, then it is also considered to be closed. In addition to its limit points, a closed set can also contain isolated points. A set is called open if each of its points is internal to it. We give examples of closed and open sets. Every segment is a closed set, and every interval is an open set. Improper half-intervals are closed, and improper intervals are open. The entire line is both a closed and an open set. It is convenient to think of the empty set as both closed and open at the same time. Any finite set of points on a line is closed, since it has no limit points. Set of points closed; this set has a single limit point that belongs to the set. Our task is to find out how an arbitrary closed or open set works. To do this, we need a number of auxiliary facts, which we will accept without proof. 1. The intersection of any number of closed sets is closed. 2. The sum of any number of open sets is an open set. 3. If a closed set is bounded from above, then it contains its upper bound. Similarly, if a closed set is bounded below, then it contains its lower bound. Let E be an arbitrary set of points on the line. We call the complement of the set E and denote by the set of all points on the line that do not belong to the set E. It is clear that if x is an external point for E, then it is an internal point for the set and vice versa. 4. If the set F is closed, then its complement is open and vice versa. Proposition 4 shows that there is a very close connection between closed and open sets: one is the complement of the other. Because of this, it suffices to study only one closed or one open set. Knowing the properties of sets of one type allows you to immediately find out the properties of sets of another type. For example, any open set is obtained by removing some closed set from a line. We proceed to the study of the properties of closed sets. We introduce one definition. Let F be a closed set. An interval having the property that none of its points belongs to the set a and the points a and do is called an adjacent interval of the set. Among adjacent intervals we will also include improper intervals or if the point a or the point belongs to the set and the intervals themselves do not intersect with F. Let us show that if a point x does not belong to a closed set, then it belongs to one of its adjacent intervals. Denote by the part of the set located to the right of the point x. Since the point x itself does not belong to the set, it can be represented in the form of an intersection Each of the sets F is closed. Therefore, by Proposition 1, the set is closed. If the set is empty, then the entire half-interval belongs to the set Let us now assume that the set is not empty. Since this set lies entirely on the half-interval, it is bounded from below. Denote by its lower bound. According to the proposal and therefore . Further, since there is an infimum of the set , then the half-interval to the left of the point does not contain points of the set and, therefore, does not contain points of the set So, we have constructed a half-interval that does not contain points of the set, and either or the point belongs to the set Similarly, a half-interval is constructed that does not contain points of the set, and either or a Now it is clear that the interval contains the point x and is an adjacent interval of the set. It is easy to see that if - two adjacent intervals of the set, then these intervals either coincide or do not intersect. It follows from the above that any closed set on the line is obtained by removing from the line a certain number of intervals, namely, adjacent intervals of the set. Since each interval contains at least one rational point, and all rational points on the line are a countable set, it is easy to verify that that the number of all adjacent intervals is more than countable. From here we get the final conclusion. Any closed set on a line is obtained by removing from the line at most a countable set of disjoint intervals. By Proposition 4, this immediately implies that any open set on the line is at most a countable sum of disjoint intervals. By virtue of Propositions 1 and 2, it is also clear that any set arranged as indicated above is indeed closed (open). As can be seen from the following example, closed sets can have a very complex structure. Cantor perfect set. Let us construct one special closed set with a number of remarkable properties. First of all, we remove the improper intervals and from the straight line. After this operation, we will have a segment. Further, we remove from this segment the interval that makes up its middle third. From each of the remaining two segments, delete its middle third. We continue this process of removing the middle thirds from the remaining segments indefinitely. The set of points on the line that remains after removing all these intervals is called the Cantor perfect set; We will refer to it as R. Let's consider some properties of this set. The set P is closed, since it is formed by removing a certain set of non-intersecting intervals from a straight line. The set P is not empty, in any case it contains the ends of all the discarded intervals. A closed set F is called perfect if it does not contain isolated points, that is, if each of its points is a limit point. Let us show that the set P is perfect. Indeed, if some point x were an isolated point of the set P, then it would serve as a common end of two adjacent intervals of this set. But, according to the construction, adjacent intervals of the set P do not have common ends. The set P does not contain any interval. Indeed, suppose that a certain interval belongs entirely to the set P. Then it belongs entirely to one of the segments obtained at the step of constructing the set P. But this is impossible, since the lengths of these segments tend to zero. It can be shown that the set P has the cardinality of the continuum. In particular, this implies that the Cantor perfect set contains, in addition to the ends of adjacent intervals, other points as well. Indeed, the ends of adjacent intervals form only a countable set. Various types of point sets are constantly encountered in the most diverse branches of mathematics, and knowledge of their properties is absolutely necessary in the study of many mathematical problems. The theory of point sets is especially important for mathematical analysis and topology. Let us give several examples of the appearance of point sets in the classical branches of analysis. Let be a continuous function given on a segment. We fix a number a and consider the set of those points x for which It is easy to show that this set can be an arbitrary closed set located on the segment. In the same way, the set of points x, for which it can be any open set If there is a sequence of continuous functions given on a segment, then the set of those points x where this sequence converges cannot be arbitrary, but belongs to a well-defined type. The mathematical discipline that studies the structure of point sets is called descriptive set theory. Very great merits in the development of descriptive set theory belong to Soviet mathematicians - N. N. Luzin and his students P. S. Aleksandrov, M. Ya. Suslin, A. N. Kolmogorov, M. A. Lavrentiev, P. S. Novikov , L. V. Keldysh, A. A. Lyapunov and others. Research by N. N. Luzin and his students showed that there is a deep connection between descriptive set theory and mathematical logic. The difficulties that arise when considering a number of problems in descriptive set theory (in particular, problems of determining the cardinality of certain sets) are difficulties of a logical nature. On the contrary, the methods of mathematical logic allow one to penetrate more deeply into some questions of descriptive set theory. Proof. 1) Indeed, if the point a belongs to the union of open sets, then it belongs to at least one of these sets, which, by the hypothesis of the theorem, is open. Hence, it belongs to some neighborhood O(a) of the point a, but then this neighborhood also belongs to the union of all open sets. Hence the point a is an internal join point. Because a is an arbitrary point of the union, then it consists only of interior points, and hence, by definition, is an open set. 2) Let now X is the intersection of a finite number of open sets . If a is a set point X, then it belongs to each of the open sets , and, therefore, is an interior point of each of the open sets. In other words, there are intervals that are entirely contained, respectively, in the sets . Denote by the smallest of the numbers . Then the interval will be contained simultaneously in all intervals , i.e. will be wholly contained in , and in ,..., and in , i.e. . From here and we conclude that any point is an interior point of the set X, i.e. a bunch of X is open. It follows from this theorem that the intersection of a finite number of neighborhoods of the point a is again a neighborhood of this point. Note that the intersection of an infinite number of open sets is not always an open set. For example, the intersection of intervals ,… is a set consisting of one point a, which is not an open set (why?). A point a is called a limit point of a set X if there is at least one point of the set X in any punctured neighborhood of this point. So, the point is the limit point of the segment ,
since in any punctured interval of a point there is a point belonging to this segment. For example, a point that satisfies the inequality . And there are obviously many such points. It is easy to prove that each point of the segment [ 0, 1]
is an marginal point of this segment. In other words, the cut
consists entirely of its limit points. A similar statement is true for any segment. Note here that all limit points of the set
belong to this section. It is also obvious that all points of the segment , will be limit points for the interval (0, 1
) (prove it!). However, there are already two limit points 0 and 1 do not belong to the interval (0, 1).
In these examples, we see that the limit points of a set may or may not belong to it. It can be proved that in any punctured neighborhood of the limit point a of the set X there are infinitely many points of the set X. A set X is called a closed set if it contains all of its limit points. So, every segment is a closed set.
Interval (0, 1)
is not a closed set, since two of its limit points do not belong to it 0 and 1. The set of all rational numbers Q is not closed, since it does not contain some of its limit points. In particular, the number is the limit point of the set Q(prove it!) but Q. Since each point of the set R is a limit point of this set and belongs to it, then R is a closed set.
Every finite set is closed, since the set of its limit points is the empty set Æ
, which belongs to the set itself. Closed sets can be bounded, for example, the segment , and unbounded, for example, the set of real numbers R. Verna
=R(2)
