On Darboux property of approximately continuous multivalued functions
by
Grazyna Kwiecinska
University of Gdansk, Poland

Let X and Y be two non-empty sets and let us assume that for every point x in X a non-empty subset F(x) of Y is given. In this case we say that F is a multivalued function from X to Y and we write F:X --> Y.

For F:X --> Y and any sets A subset X and B subset Y we denote

F⁺(B)={x in X:F(x) subset B} and F^-(B)={x in X:F(x) \cap B =/= \emptyset}.

Let (X, T(X)) and (Y, T(Y)) be two topological spaces. A multivalued function F:X --> Y is called upper (resp. lower) semicontinuous if for every G in T(Y) the set F⁺(G) (resp. F^-(G)) is open in X.

F is called continuous if it is both upper and lower semicontinuous.

Let R denotes the set of real numbers and let I subset R denotes an interval contained in R. Let F:I --> R be a multivalued function. In the work [3] the following definition of a Darboux property was given: F has the property D if the image F(E) is connected for any connected set E subset I. Another condition equivalent to the D property of F is that F takes closed intervals into connected sets.

Continuous multivalued functions not necessarily have the property D, but the following is true. ([2], theorem 2) If a multivalued function F:I --> R with closed and connected values is continuous, then it takes connected sets into connected sets.

Let (X, d, M(X), \mu) be a separable metric space with a metric d, with a \sigma-finite G_\delta-regular complete measure \mu defined on a \sigma-field M(X) of subsets of X containing Borel sets. Let \mu^* be the outer measure corresponding to \mu.

Let N(X) subset M(X) be a net, i.e. N(X) is a denumerable disjoint family of M(X)-measurable sets (called cells) covering X with a positive and finite measure \mu, the boundaries of which are of \mu-measure zero. Let \bigtriangleup(N(X)) denotes upper bound of the diameters \delta(I) of all cells I of the net N(X).

Let (N_n(X))_{n in N} be a regular sequence of nets, i.e. each cell of the net N_n+1(X) is a subset of some cell of N_n(X) and \bigtriangleup(N_n(X)) --> 0 as n --> \infty.

Finally let F= \cup _{n in N}N_n(X). The family F forms a net structure of (X, d, M(X), \mu) in accordance with Bruckner's terminology ([1], 6.3, p.35).

If x in X, then for each n in N there exists exactly one cell I_n of N_n(X) containing x. We take I_n --> x to mean that x in I_n in N_n(X) and n --> \infty.

Let A subset X and x in X. The upper outer density of A at the point x with respect to F is

limsupIn --> x  \mu*(A \cap In) \mu(In) .

Replacing limsup by liminf we obtain the lower outer density of A at x with respect to F. These densities we will denote by D^*_u(x, A) and D^*_l(x, A) respectively. If both these densities are equal, then their common value is called the outer density of the set A at the point x with respect to F and is denoted by D^*(x, A).

If A in M(X), then outer density of the set A at the point x in X with respect to F is called density of A at x with respect to F.

A point x in X is called density point of a set A subset X with respect to F if there exists a set B in M(X) such that B subset A and density of B at x with respect to F is equal to 1. We will write D(x, A)=1.

Let us assume that

1 F has the density property, i.e.

for allA subset X \mu({x in A : D^*_l(x, A) < 1}) = 0.

A measurable set in X is called homogeneous with respect to F if its density with respect to F is one at each of its point.

The space X can be topologized by taking the homogeneous sets with respect to F as open sets. This topology we will denote by T_d(X) (for more details see [4] or [5]).

Let (Y, T(Y)) be a topological space. Let F:X --> Y be a multivalued function and assume that x₀ in X. F is called approximately upper (resp. lower) semicontinuous at the point x₀ with respect to F if there exists a set A in M(X) such that D(x₀, A)=1 and the restriction F| _A is upper (resp. lower) semicontinuous at x₀.

F is called approximately upper (resp. lower) semicontinuous with respect to F if it is approximately upper (resp. lower) semicontinuous with respect to F at each point x in X.

If F is simultaneously approximately upper semicontinuous and approximately lower semicontinuous with respect to F, then it is called approximately continuous with respect to F. Now let (Y, U(Y) be a uniform space. ([6], theorem 4) Let F:X --> Y be a multivalued function with compact values. Then F is approximately continuous if and only if F is continuous with respect to the density topology T_d(X). Let A subset X and x in X. Let us note that

2 x is a limit point of the set A in the topology T_d(X) if and only if D_u^*(x, A) > 0.

Let A subset X be a closed set with connected interior. The set A is called T_d(X)-regular if its boundary points are T_d(X)-limit points of the interior of A. If a set A subset X is T_d(X)-regular, then it is T_d(X)-connected.

Let F:X --> Y be an approximately continuous with respect to F multivalued function with closed and connected values. Then F takes T_d(X)-regular sets into connected sets, i.e. it has a Darboux property.

#1Bruckner A. M., Differentiation of integrals", Amer. Math. Monthly 78, (1971), P II, pp.1-54.

#1Czarnowska J., Kwieci\'nska G., Ön the Darboux property of multivalued functions", Demonstr. Math. XXV (1992), No.1-2, pp. 192-199.

#1Ewert J., Lipi\'nski J., Ön the continuity of Darboux multifunctions", Real Anal. Ex. 13, No 1 (1987-88), pp. 122 - 125.

#1Franklin D. Tall, "The density topology", Pacific Journal of Math., vol. 62, No 1 (1976), pp. 275-284.

#1Goffman C., Neugebauer C., Nishura T., "Density topology and approximate continuity", Duke Math. J., 28, (1961), pp.497-506.

#1Kwieci\'nska G., Äpproximately continuous multivalued functions", Proc. Eight Prague Topol. Symp, pp. 135-140.

Date received: June 30, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-52.