Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications. Valeri Obukhovskii

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

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.