Читать книгу Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications - Valeri Obukhovskii - Страница 8

In order to know something you first need to know something.

—Stanislaw Lem

This chapter contains the preliminary information mainly from the general topology which is necessary for the further reading (details may be found, for example, in [1], [131], [135], [241], [270]). A reader familiar with these topics can pass on directly to Chapter 1.

We will use standard symbols x ∈ X (x ∉ X), X ⊂ Y, X = Y \ X to denote the belonging (not belonging) of an element to a set, the inclusion of a set into a set (notice that the symbol ⊂ does not exclude the equality of sets X and Y) and the complement of a set X with respect to the whole space Y. If is a certain family of sets, by the symbols and we denote the union and, respectively, the intersection of the sets of this family. The Cartesian product of sets X and Y is denoted by X × Y. By the symbol {x|M(x)}, the set of objects x possessing a property M(x) is denoted.

Let X be a set; by a topology on X we mean a system τ of subsets of X satisfying the following conditions:

1) and X belong to τ;

2)if U and V belong to τ then their intersection U ⋂ V belongs to τ;

3)the union of each family of sets from τ belongs to τ.

Elements of the system τ are called open sets, and the pair (X, τ) is said to be a topological space. If the topology τ is implicitly meant we say simply about the topological space X.

If τ is a topology on a space X then its base is a subsystem τ₁ ⊂ τ such that each element from τ can be represented as the union of a certain family of elements from τ₁.

The set of real numbers usually is endowed with the topology whose base consists of intervals (a, b).

Let (X, τ) be a topological space; a neighborhood of a point x ∈ X is any subset of X in which an open set containing x lies. Similarly, any subset of X in which an open set containing a subset A ⊂ X lies is called a neighborhood of a set A. A subset of X is open if and only if it is a neighborhood for each of its points.

The interior of a set A is the largest open set contained in A, it will be denoted as int A. The points of int A are called interior points of A.

A subset A of a topological space X is called closed if its complement X \ A is open. The closure of a set A is the least closed set containing A. A set is closed if and only if it coincides with its closure.

A topological space X is said to be separable, if it contains a countable subset A which is dense in X, i.e., = X.

Let (X₁, τ₁), (X₂, τ₂) be topological spaces; the topology in the Cartesian product X₁ × X₂ is generated in the following way: its base consists of the sets having the form U_α × U_β, where U_α ∈ τ₁, Uβ ∈ τ₂. The set X₁ × X₂ endowed with such topology is called the topological product of X₁ and X₂.

A topological space X is called:

(i)T₁-space if each one-point set in X is closed;

(ii)Hausdorff if any two different points from X possess disjoint neighborhoods;

(iii)regular if it is a T₁-space such that any point from X and any closed subset to which it does not belong, possess disjoint neighborhoods;

(iv)normal if it is a T₁-space such that any two its disjoint closed subsets possess disjoint neighborhoods.

Let X, Y be topological spaces; a map f : X → Y is continuous if the set f⁻¹(V) = {x|x ∈ X, f(x) ∈ V} is open in X for each open set V ⊂ Y.

A topological space X defines on each of its subsets A ⊂ X the topology whose open sets are the intersections of A with open subsets of X. Such topology is called relative or induced. The subset A with this topology is said to be a subspace (of a space X). If A is a subspace of X then a map i : A → X defined by the rule i (x) = x is called the inclusion map. It is easy to see that the inclusion map is continuous.

A subset A of a topological space X is called connected, if it can not be represented as the union of two nonempty disjoint open (in the relative topology) sets. An open connected subset of a topological space is said to be a domain.

Let X be a topological space; a function f : X → is called upper [lower] semicontinuous at a point x ∈ X if for every ε > 0 there exists a neighborhood U (x) of the point x such that f(x′) < f(x) + ε for all x′ ∈ U(x) [respectively, f(x′) > f(x) → ε for all x ∈ U(x)]. If a function f is upper [or lower] semicontinuous at each point of a space X it is called upper [or, respectively, lower] semicontinuous. It is easy to see that a function f is upper [lower] semicontinuous if for every r ∈ the set

respectively,

is open. While considering upper semicontinuous functions, it is often convenient to assume that they act into the extended set of real numbers obtained from by addition of +∞ and −∞.

Let be a set with a given binary relation ≤. The set is called directed if the following conditions hold:

1)α ≤ β, β ≤ γ imply α ≤ γ for every α, β, γ ∈ ;

2)α ≤ α for every α ∈ ;

3)for every α, β ∈ there exists γ ∈ such that α ≤ γ, β ≤ γ.

A map of a directed set into a topological space X, i.e., the correspondence which assigns to each α ∈ a certain x_α ∈ X is said to be a net or a generalized sequence.

A net {x_α} ⊂ X converges to a point x ∈ X if for every neighborhood U of the point x there exists an index α₀ such that x_α ∈ U for all α ≥ α₀. A point x belongs to the closure of a subset M of a space X if and only if M contains a net converging to x. In the case of a metric space X a net here may be substituted with a usual sequence.

A cover of a set X is a collection Σ of subsets of X whose union is the whole X. A cover ∑′ is called subcover of a cover Σ provided each of sets from the collection ∑′ belongs to Σ. If each cover of a topological space X by open sets contains a finite subcover then the space X is called compact.

The compactness of a space X is equivalent to each of the following conditions:

1)each net in X contains a convergent subnet;

2)each centered collection of closed subsets of X (i.e., such collection that each its nonempty finite subcollection has a nonempty intersection) also has a nonempty intersection.

A set X is said to be relatively compact, if its closure is compact. A upper semicontinuous function f : X → defined on a compact space X reaches its maximum, whereas a lower semicontinuous function reaches its minimum.

By virtue of the Tychonoff theorem the topological product X₁ × X₂ of compact spaces X₁ and X₂ is compact.

A subset of the Euclidean n-dimensional space ⁿ is relatively compact if and only if it is bounded.

A cover Σ of a topological space X is called locally finite if every point x ∈ X possesses a neighborhood U which intersects only a finite number of sets from Σ. A topological space X is said to be paracompact if it is Hausdorff and each its open cover Δ has an open locally finite refinement Σ (i.e., each of the sets from Σ is contained in a set from Δ).

For each locally finite open cover Ξ = {Ui}_j∈J of a paracompact space X there exists a subordinated partition of unity, i.e., a family {pj}_j∈J of continuous on X nonnegative functions such that:

1)for each j ∈ J we have: supp pj = {x|x ∈ X, pj(x) ≠ 0} ⊂ U_j;

2)for each point .

Notice that due to the local finiteness of the cover Ξ, only a finite number of terms in the last sum differ from zero.

Let (X, ϱ) be a metric space, x ∈ X, and r > 0. The set

is called an open ball of the radius r with the center at x, whereas the set

is a closed ball of the radius r with the center at x. The collection of all open balls is the base of a certain topology on X which is called metric topology. It is clear that a set V in a metric space X is open if and only if every point x of V belongs to V with a certain open ball centered at x.

Two metrics on a set X are called equivalent if they generate on X the same metric topology. Each space with metric topology is normal and hence regular and Hausdorff. Due to the Stone theorem every metric space is paracompact.

The distance from a point x to a set A ⊂ X is defined as

If A ⊂ X and ε > 0 then the set

is called an ε-neighborhood of the set A.

Let A be a compact metric space and Σ an open cover of X. Then, according to the Lebesgue covering lemma (see, e.g., [241]), there exists a positive real number r with the property that for each x ∈ X there exists a set U ∈ Σ such that B_r(x) ⊂ U. From this assertion it follows that if (X, ϱ) is a metric space, A is a compact subset of X, B is a closed subset of X and A ⋂ B = then

and hence there exists such ε > 0 that

This yields, in particular, that for every open neighborhood U of a compact set A there exists a sufficiently small ε > 0 such that the ε-neighborhood of A is contained in U.

Let T be a compact space, (X, ϱ) a metric space. On the set C(T, X) of all continuous functions from T to X the matric may be defined by the formula

The topology τ_c generated on C(T, X) by this metric is called the topology of uniform convergence.

Let (T, ϱ_T) be a compact metric space. A family of functions H ⊂ C(T, X) is called equicontinuous if for each ε > 0 there exists δ > 0 such that for every t, t′ ∈ T the condition ϱT(t, t′) < δ implies ϱ(f(t), f(t′)) < ε for all f ∈ H. According to the Arzela–Ascoli theorem (see, e.g., [241]) if a subset H ⊂ C(T, X) is equicontinuous and the sets H(t) = {f(t) |f ∈ H} are relatively compact in X for all t ∈ T then H is relatively compact in the space (C(T, X), τ_c).

If X is a linear space and A, B ⊂ X then

If α ∈ then

The set of all finite linear combinations

where and every x_i belongs to A is a least convex set containing A and it is called the convex hull of the set A and is denoted as coA.

Let X be a linear space on which a topology τ be defined. The pair (X, τ) is said to be a linear topological space if the topology τ is consistent with the linear operations on X in the following way: 1) the addition operation is continuous, i.e., the map

is continuous; 2) the number multiplication operation is continuous, i.e., the map

is continuous.

If X is a linear topological space and A ⊂ X then the closure of the set coA is denoted by . It is called the convex closure A and it is a least convex closed set containing A.

The following Brouwer fixed point theorem holds true. If M is a convex closed subset of a finite-dimensional linear topological space then every continuous map f : M → M such that its range f(M) is bounded has at least one fixed point x ∈ M, x = f(x).

We will suppose that the reader is familiar with the concepts of normed and Banach spaces as well as with a main information concerning their properties (see, e.g., [120], [124], [247], [371], [384] and others).

Nevertheless, let us indicate the following facts that we will use in the sequel.

Let A be a closed subset of a metric space X and Y a normed space. Then each continuous map f : A → Y has a continuous extension : X → Y and, moreover (the Tietze-Dugundji theorem). The following assertion is an immediate consequence of this result. If Y is a normed space and A is its nonempty closed convex subset then there exists a continuous map (the retraction) r : Y → A such that r(y) = y for all y ∈ A.

If X is a Banach space and A ⊂ X is a compact set then its convex closure is also compact (the Mazur theorem).

If A is a bounded subset of a normed space X then by the norm of the set A we mean the value

We will assume also that the reader is familiar with the notions of the Lebesgue measure, a measurable and a Bochner integrable function with the values in a Banach space as well as with main properties of the space of integrable functions L¹(see, e.g., [124], [316], [371], [384], [408]).

The sign := will denote the equality by definition.

The end of the proof will be marked with the symbol ■.