Open and closed sets. Metric spaces and continuous mappings

One of the main tasks of the theory of point sets is the study of properties various types point sets. Let's get acquainted with this theory using two examples and study the properties of the so-called closed and open sets.

The set is called closed , if it contains all its limit points. If a set does not have a single limit point, then it is also considered 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 for it.

Let's give 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 consider the empty set to be both closed and open at the same time. Any finite set points on a line is closed, since it has no limit points.

A set consisting of points:

closed; this set has a unique limit point x=0, which belongs to the set.

The main task is to find out how an arbitrary closed or open set is structured. To do this, we will 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 above, then it contains its own top edge. Similarly, if a closed set is bounded below, then it contains its infimum.

Let E be an arbitrary set of points on a line. We call the complement of the set E and denote by CE the set of all points on the line not belonging to many 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 a set F is closed, then its complement CF is open and vice versa.

Proposition 4 shows that there is quite a difference between closed and open sets. close connection: some are complements of others. Because of this, it is sufficient to study only closed or only open sets. 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.

Let's start studying the properties of closed sets. Let's introduce one definition. Let F be a closed set. An interval (a, b) having the property that none of its points belong to the set F, but the points a and b belong to F, is called an adjacent interval of the set F.

We will also include improper intervals among adjacent intervals, or if point a or 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.

Let us 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 intersection form:

Each of the sets is F and 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 is entirely located on a half-interval, it is bounded below. Let us denote its lower bound by b. According to Proposition 3, which means. Further, since b is the infimum of the set, 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. So, we have constructed a half-interval (x, b) not containing 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.

From the previous it follows that any closed set on a line is obtained by removing a certain number of intervals from the line, namely adjacent intervals of the set F. Since each interval contains at least one rational point, and there is a countable set of all rational points on the line, it is easy make sure that the number of all adjacent intervals is at most countable. From here we get the final conclusion. Every closed set on a line is obtained by removing from the line at most a countable set of disjoint intervals.

By virtue of Proposition 4, it immediately follows that every open set on a line is nothing more than 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. Let's get acquainted with this theory using two examples and study the properties of the so-called closed and open sets.

A set is called closed if it contains all its limit points. If a set does not have a single limit point, then it is also considered 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.

Let us give examples of closed and open sets. Every segment \(\) is a closed set, and every interval \((a,b)\) is an open set. Improper half-intervals \((-\infty,b]\) and \(\) , not containing points of the set \(F\) , and either \(a=-\infty\) or \(a\in F\) . 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 \((a_1,b_1)\) and \( (a_2,b_2)\) are two adjacent intervals of the set \(F\), then these intervals either coincide or do not intersect.

From the previous it follows that any closed set on a line is obtained by removing a certain number of intervals from the line, namely adjacent intervals of the set \(F\) . Since each interval contains at least one rational point, and there is a countable set of all rational points on a line, it is easy to verify that the number of all adjacent intervals is at most countable. From here we get the final conclusion. Every closed set on a line is obtained by removing from the line at most a countable set of disjoint intervals.

By virtue of Proposition 4, it immediately follows that every open set on a line is nothing more than 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 the series remarkable properties. First of all, let’s remove the improper intervals \((-\infty,0)\) and \((1,+\infty)\) from the line. After this operation we will be left with the segment \(\) . Next, remove the interval from this segment \(\left(\frac(1)(3),\frac(2)(3)\right)\), making up its middle third. From each of the remaining two segments \(\left\) And \(\left[\frac(2)(3),1\right]\) Let's remove its middle third. We will continue this process of removing the middle thirds of the remaining segments indefinitely. The set of points on the line remaining after removing all these intervals is called the Cantor perfect set; we will denote it by the letter \(P\) .

Let's consider some properties of this set. The set \(P\) is closed, since it is formed by removing from the line a certain set of disjoint intervals. The set \(P\) is not empty; in any case, it contains the ends of all discarded intervals.

The closed set \(P\) 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 the 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 a single interval. In fact, let us assume that some interval \(\delta\) entirely belongs to the set \(P\) . Then it entirely belongs to one of the segments obtained at the \(n\) -th step of constructing the set \(P\) . But this is impossible, since at \(n\to\infty\) the lengths of these segments tend to zero.

It can be shown that the set \(P\) has the cardinality of a continuum. In particular, it follows that the Cantor perfect set contains, in addition to the ends of adjacent intervals, other points. Indeed, the ends of adjacent intervals form only a countable set.

Various types of point sets are constantly encountered in various branches of mathematics, and knowledge of their properties is absolutely necessary when studying many mathematical problems. Especially great value has point set theory for mathematical analysis and topology.

Let us give several examples of the appearance of point sets in classical sections of analysis. Let \(f(x)\) be a continuous function defined on the interval \(\) . Let us fix the number \(\alpha\) and consider the set of those points \(x\) for which \(f(x)\geqslant\alpha\) . 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 \(f(x)>\alpha\) can be any open set \(G\subset\) . If \(f_1(x),f_2(x),\ldots,f_n(x),\ldots\) there is a sequence continuous functions, given on the segment \(\), then the set of those points \(x\) where this sequence converges cannot be arbitrary, but belongs to a very specific type.

The mathematical discipline that studies the structure of point sets is called descriptive set theory. Very great achievements in the development of descriptive set theory belong to Soviet mathematicians- N. N. Luzin and his students P. S. Alexandrov, M. Ya. Suslin, A. N. Kolmogorov, M. A. Lavrentiev, P. S. Novikov, L. V. Keldysh, A. A. Lyapunov etc.

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 of descriptive set theory (in particular, problems of determining the cardinality of certain sets) are difficulties of a logical nature. On the contrary, methods mathematical logic allow us to penetrate more deeply into some questions of descriptive set theory.

Open and closed sets

Appendix 1 . Open and closed sets

Many M on a straight line is called open, if each of its points is contained in this set along with a certain interval. Closed is a set that contains all its limit points (i.e., such that any interval containing this point intersects the set at least at one more point). For example, a segment is a closed set, but is not open, and an interval, on the contrary, is an open set, but is not closed. There are sets that are neither open nor closed (for example, a half-interval). There are two sets that are both closed and open - this is empty and that's it Z(prove that there are no others). It's easy to see that if M open, then [` M] (or Z \ M- addition to set M to Z) is closed. Indeed, if [` M] is not closed, then it does not contain any limit point of its own m. But then m ABOUT M, and each interval containing m, intersects with the set [` M], i.e. has a point not lying in M, and this contradicts the fact that M– open. Similarly, also directly from the definition, it is proved that if M is closed, then [` M] open (check!).

Now we will prove the following important theorem.

Theorem. Any open set M can be represented as a union of intervals with rational ends (that is, with ends at rational points).

Proof . Consider the union U all intervals with rational ends that are subsets of our set. Let us prove that this union coincides with the entire set. Indeed, if m- some point from M, then there is an interval ( m 1 , m 2) M M containing m(this follows from the fact that M– open). On any interval you can find a rational point. Let on ( m 1 , m) - This m 3, on ( m, m 2) – this is m 4. Then point m covered by union U, namely, the interval ( m 3 , m 4). Thus, we have proven that each point m from M covered by union U. In addition, as it obviously follows from the construction U, no point not contained in M, not covered U. Means, U And M match.

An important consequence of this theorem is the fact that any open set is countable combining intervals.

Nowhere dense sets and sets of measure zero. Cantor set>

Appendix 2 . Nowhere dense sets and sets of measure zero. Cantor set

Many A called nowhere dense, if for any different points a And b there is a segment [ c, d] M [ a, b], not intersecting with A. For example, the set of points in the sequence a n = [ 1/(n)] is nowhere dense, but a set rational numbers- No.

Baire's theorem. A segment cannot be represented as a countable union of nowhere dense sets.

Proof . Suppose there is a sequence A k nowhere dense sets such that And i A i = [a, b]. Let's construct the following sequence of segments. Let I 1 – some segment embedded in [ a, b] and does not intersect with A 1. By definition, a nowhere dense set on an interval I 1 there is a segment that does not intersect with the set A 2. Let's call him I 2. Further, on the segment I 2, similarly take the segment I 3, not intersecting with A 3, etc. Sequence I k there are nested segments common point(this is one of the main properties real numbers). By construction, this point does not lie in any of the sets A k, which means that these sets do not cover the entire segment [ a, b].

Let's call the set M having measure zero, if for any positive e there is a sequence I k intervals with a total length less than e, covering M. Obviously, any countable set has measure zero. However, there are also uncountable sets that have measure zero. Let's build one, very famous, called Cantor's.

Let's take a segment. Let's divide it into three equal parts. Let's throw out the middle segment (Fig. 11, A). There will be two segments of total length [2/3]. We will perform exactly the same operation with each of them (Fig. 11, b). There will be four segments left with total length [ 4/9] = ([ 2/3]) \ B 2 . Continuing like this (Fig. 11, V–e) to infinity, we obtain a set that has a measure less than any predetermined positive measure, i.e., measure zero. It is possible to establish a one-to-one correspondence between the points of this set and infinite sequences of zeros and ones. If during the first “throwing out” our point falls into the right segment, we will put 1 at the beginning of the sequence, if in the left - 0 (Fig. 11, A). Next, after the first “throwing out”, we get a small copy of the large segment, with which we do the same thing: if our point after throwing out falls into the right segment, we put 1, if it’s in the left one – 0, etc. (check the one-to-one relationship) , rice. 11, b, V. Since the set of sequences of zeros and ones has cardinality continuum, the Cantor set also has cardinality continuum. Moreover, it is easy to prove that it is not dense anywhere. However, it is not true that it has strict measure zero (see the definition of strict measure). The idea of ​​proving this fact is as follows: take the sequence a n, tending to zero very quickly. For example, the sequence a n = [ 1/(2 2 n)]. Then we will prove that this sequence cannot cover the Cantor set (do it!).

Appendix 3 . Tasks

Set Operations

Sets A And B are called equal, if each element of the set A belongs to many B, and vice versa. Designation: A = B.

Many A called subset sets B, if each element of the set A belongs to many B. Designation: A M B.

1. For each two of the following sets, indicate whether one is a subset of the other:

2. Prove that the set A if and only if is a subset of the set B, when every element not belonging to B, does not belong A.

3. Prove that for arbitrary sets A, B And C

A) A M A; b) if A M B And B M C, That A M C;

V) A = B, if and only if A M B And B M A.

The set is called empty, if it does not contain any elements. Designation: F.

4. How many elements does each of the following sets have:

5. How many subsets does a set of three elements have?

6. Can a set have exactly a) 0; b*) 7; c) 16 subsets?

Association sets A And B x, What x ABOUT A or x ABOUT B. Designation: A AND B.

By crossing sets A And B is called a set consisting of such x, What x ABOUT A And x ABOUT B. Designation: A Z B.

By difference sets A And B is called a set consisting of such x, What x ABOUT A And x P B. Designation: A \ B.

7. Given sets A = {1,3,7,137}, B = {3,7,23}, C = {0,1,3, 23}, D= (0,7,23,1998). Find the sets:

8. Let A is the set of even numbers, and B– set of numbers divisible by 3. Find A Z B.

9. Prove that for any sets A, B, C

A) A AND B = B AND A, A Z B = B Z A;

b) A AND ( B AND C) = (A AND B)AND C, A Z ( B Z C) = (A Z B)Z C;

V) A Z ( B AND C) = (A Z B)AND ( A Z C), A AND ( B Z C) = (A AND B)Z ( A AND C);

G) A \ (B AND C) = (A \ B)Z ( A \ C), A \ (B Z C) = (A \ B)AND ( A \ C).

10. Is it true that for any sets A, B, C

Set mappings

If each element x sets X exactly one element is matched f(x) sets Y, then they say that it is given display f from many X into the multitude Y. At the same time, if f(x) = y, then the element y called way element x when displayed f, and the element x called prototype element y when displayed f. Designation: f: X ® Y.

11. Draw all possible mappings from the set (7,8,9) to the set (0,1).

Let f: X ® Y, y ABOUT Y, A M X, B M Y. Full prototype of the element y when displayed f is called a set ( x ABOUT X | f(x) = y). Designation: f - 1 (y). In the image of the multitude A M X when displayed f is called a set ( f(x) | x ABOUT A). Designation: f(A). The prototype of the set B M Y is called a set ( x ABOUT X | f(x) ABOUT B). Designation: f - 1 (B).

12. To display f: (0,1,3,4) ® (2,5,7,18), given by the picture, find f({0,3}), f({1,3,4}), f - 1 (2), f - 1 ({2,5}), f - 1 ({5,18}).

13. Let f: X ® Y, A 1 , A 2 M X, B 1 , B 2 M Y. Is it always true that

A) f(X) = Y;

b) f - 1 (Y) = X;

V) f(A 1 I A 2) = f(A 1)And f(A 2);

G) f(A 1 W A 2) = f(A 1)Z f(A 2);

d) f - 1 (B 1 I B 2) = f - 1 (B 1)And f - 1 (B 2);

e) f - 1 (B 1 W B 2) = f - 1 (B 1)Z f - 1 (B 2);

g) if f(A 1) M f(A 2), then A 1 M A 2 ;

h) if f - 1 (B 1) M f - 1 (B 2), then B 1 M B 2 ?

Composition mappings f: X ® Y And g: Y ® Z is called a mapping that associates an element x sets X element g(f(x)) sets Z. Designation: g° f.

14. Prove that for arbitrary mappings f: X ® Y, g: Y ® Z And h: Z ® W the following is done: h° ( g° f) = (h° g)° f.

15. Let f: (1,2,3,5) ® (0,1,2), g: (0,1,2) ® (3,7,37,137), h: (3,7,37,137) ® (1,2,3,5) – mappings shown in the figure:

Draw pictures for the following displays:

A) g° f; b) h° g; V) f° h° g; G) g° h° f.

Display f: X ® Y called bijective, if for each y ABOUT Y there is exactly one x ABOUT X such that f(x) = y.

16. Let f: X ® Y, g: Y ® Z. Is it true that if f And g are bijective, then g° f bijectively?

17. Let f: (1,2,3) ® (1,2,3), g: (1,2,3) ® (1,2,3), – mappings shown in the figure:

18. For each two of the following sets, find out whether there is a bijection from the first to the second (assuming that zero is a natural number):

a) many natural numbers;

b) the set of even natural numbers;

c) the set of natural numbers without the number 3.

Metric space called a set X with a given metric r: X× X ® Z

1) " x,y ABOUT X r ( x,y) i 0, and r ( x,y) = 0 if and only if x = y (non-negativity ); 2) " x,y ABOUT X r ( x,y) = r ( y,x) (symmetry ); 3) " x,y,z ABOUT X r ( x,y) + r ( y,z) i r ( x,z) (triangle inequality ). 19 19. X

A) X = Z, r ( x,y) = | x - y| ;

b) X = Z 2 , r 2 (( x 1 ,y 1),(x 2 ,y 2)) = C (( x 1 - x 2) 2 + (y 1 - y 2) 2 };

V) X = C[a,ba,b] functions,

Open(respectively, closed) ball of radius r in space X centered at a point x called a set U r (x) = {y ABOUT x:r ( x,y) < r) (respectively, B r (x) = {y ABOUT X:r ( x,y) Ј r}).

Internal point sets U M X U

open surroundings this point.

Limit point sets F M X F.

closed

20. Prove that

21. Prove that

b) union of a set A short circuit A

Display f: X ® Y called continuous

22.

23. Prove that

F (x) = inf y ABOUT F r ( x,y

F.

24. Let f: X ® Y– . Is it true that its inverse is continuous?

Continuous one-to-one mapping f: X ® Y homeomorphism. Spaces X, Yhomeomorphic.

25.

26. For which couples? X, Y f: X ® Y, which does not stick together points (i.e. f(x) № f(y) at x № y investments)?

27*. local homeomorphism(i.e. at each point x plane and f(x) torus there are such neighborhoods U And V, What f homeomorphically maps U on V).

Metric spaces and continuous mappings

Metric space called a set X with a given metric r: X× X ® Z, satisfying the following axioms:

1) " x,y ABOUT X r ( x,y) i 0, and r ( x,y) = 0 if and only if x = y (non-negativity ); 2) " x,y ABOUT X r ( x,y) = r ( y,x) (symmetry ); 3) " x,y,z ABOUT X r ( x,y) + r ( y,z) i r ( x,z) (triangle inequality ). 28. Prove that the following pairs ( X,r ) are metric spaces:

A) X = Z, r ( x,y) = | x - y| ;

b) X = Z 2 , r 2 (( x 1 ,y 1),(x 2 ,y 2)) = C (( x 1 - x 2) 2 + (y 1 - y 2) 2 };

V) X = C[a,b] – set of continuous on [ a,b] functions,

Open(respectively, closed) ball of radius r in space X centered at a point x called a set U r (x) = {y ABOUT x:r ( x,y) < r) (respectively, B r (x) = {y ABOUT X:r ( x,y) Ј r}).

Internal point sets U M X is a point that is contained in U together with some ball of non-zero radius.

A set all of whose points are interior is called open. An open set containing this point, called surroundings this point.

Limit point sets F M X is a point such that any neighborhood of which contains infinitely many points of the set F.

A set that contains all its limit points is called closed(compare this definition with the one given in Appendix 1).

29. Prove that

a) a set is open if and only if its complement is closed;

b) the finite union and countable intersection of closed sets is closed;

c) the countable union and finite intersection of open sets are open.

30. Prove that

a) the set of limit points of any set is a closed set;

b) union of a set A and the set of its limit points ( short circuit A) is a closed set.

Display f: X ® Y called continuous, if the prototype of each open set open.

31. Prove that this definition is consistent with the definition of continuity of functions on a line.

32. Prove that

a) distance to set r F (x) = inf y ABOUT F r ( x,y) is a continuous function;

b) the set of zeros of the function in point a) coincides with the closure F.

33. Let f: X ® Y

Continuous one-to-one mapping f: X ® Y, the inverse of which is also continuous is called homeomorphism. Spaces X, Y, for which such a mapping exists, are called homeomorphic.

34. For each pair of the following sets, determine whether they are homeomorphic:

35. For which couples? X, Y spaces from the previous problem there is a continuous mapping f: X ® Y, which does not stick together points (i.e. f(x) № f(y) at x № y– such mappings are called investments)?

36*. Come up with a continuous mapping from a plane to a torus that would be local homeomorphism(i.e. at each point x plane and f(x) torus there are such neighborhoods U And V, What f homeomorphically maps U on V).

Completeness. Baire's theorem

Let X– metric space. Subsequence x n its elements are called fundamental, If

37. Prove that the convergent sequence is fundamental. Is the opposite statement true?

The metric space is called complete, if any fundamental sequence it converges.

38. Is it true that a space homeomorphic to a complete one is complete?

39. Prove that a closed subspace of a complete space is itself complete; the complete subspace of an arbitrary space is closed in it.

40. Prove that in a complete metric space a sequence of nested closed balls with radii tending to zero has a common element.

41. Is it possible in previous task remove the condition of completeness of space or the tendency of the radii of balls to zero?

Display f metric space X called into oneself compressive, If

42. Prove that the contraction map is continuous.

43. a) Prove that a contraction mapping of a complete metric space into itself has exactly one fixed point.

b) Place a map of Russia at a scale of 1:20,000,000 on a map of Russia at a scale of 1:5,000,000. Prove that there is a point whose images on both maps coincide.

44*. Is there an incomplete metric space in which the statement of the problem is true?

A subset of a metric space is called dense everywhere, if its closure coincides with the entire space; nowhere dense– if its closure does not have non-empty open subsets (compare this definition with the one given in Appendix 2).

45. a) Let a, b, a , b O Z And a < a < b < b. Prove that the set of continuous functions on [ a,b], monotone on , nowhere dense in the space of all continuous functions on [ a,b] with uniform metric.

b) Let a, b, c, e O Z And a < b, c> 0, e > 0. Then the set of continuous functions on [ a,b], such that

46. (Generalized Baire's theorem .) Prove that a complete metric space cannot be represented as the union of a countable number of nowhere dense sets.

47. Prove that the set of continuous, non-monotone on any non-empty interval and nowhere differentiable functions defined on the interval is everywhere dense in the space of all continuous functions on with a uniform metric.

48*. Let f– differentiable function on the interval. Prove that its derivative is continuous everywhere dense set points. This is the definition Lebesgue measures zero. If the countable number of intervals is replaced by a finite one, we get the definition Jordanova measures zero.

DEFINITION 5. Let X be a metric space, ММ Х, аОХ. A point a is called a limit point of M if in any neighborhood of a there are points of the set M\(a). The latter means that in any neighborhood of a there are points of the set M different from a.

Notes. 1. A limit point may or may not belong to the set. For example, 0 and 1 are limit points of the set (0,2), but the first does not belong to it, and the second does.

2. A point of a set M may not be its limit point. In this case, it is called an isolated point M. For example, 1 - isolated point sets (-1,0)È(1).

3. If the limit point a does not belong to the set M, then there is a sequence of points x n ОM converging to a in this metric space. To prove it, it is enough to take open balls at this point of radii 1/n and select from each ball a point belonging to M. The converse is also true, if for a there is such a sequence, then the point is a limit point.

DEFINITION 6. The closure of a set M is the union of M with the set of its limit points. Designation

Note that the closure of a ball does not have to coincide with a closed ball of the same radius. For example, in a discrete space, the closure of the ball B(a,1) is equal to the ball itself (consists of one point a) while the closed ball (a,1) coincides with the entire space.

Let us describe some properties of the closure of sets.

1. MÌ. This follows directly from the definition of a closure.

2. If M М N, then М . Indeed, if a О , a ПМ, then in any neighborhood of a there are points of the set M. They are also points of N. Therefore aО . For points from M this is clear by definition.

4. .

5. The closure of an empty set is empty. This agreement does not follow from general definition, but is natural.

DEFINITION 7. A set M М X is called closed if = M.

A set M М X is called open if the set X\M is closed.

A set M М X is said to be everywhere dense in X if = X.

DEFINITION 8. A point a is called an interior point of the set M if B(a,r)МM for some positive r, i.e. internal point is included in the set together with some neighborhood. A point a is called an exterior point of the set M if the ball B(a,r)МХ/M for some positive r, i.e., the interior point is not included in the set along with some neighborhood. Points that are neither interior nor exterior points of the set M are called boundary points.

Thus, boundary points are characterized by the fact that in each of their neighborhoods there are points both included and not included in M.

PROPOSITION 4. In order for a set to be open, it is necessary and sufficient that all its points are interior.

Examples of closed sets on a line are , )