Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

ON LOWER SEMICONTINUOUS MULTIFUNCTIONS IN QUASI-UNIFORM AND VECTOR SPACES

ANDRZEJ SPAKOWSKI

Abstract. Given a cover B of a quasi-uniform space Y we introduce a concept of lower semicontinuity for multifunctions F : X → 2^{Y}, called B-lsc. In this way, we get a common description of Vietoris-lsc, Hausdorff-lsc, and bounded-Hausdorff-lsc as well. Further, we examine set-theoretical and vector operations on such multifunctions. We also point out that the convex hull of Hausdorff-lsc multifunctions need not to be Hausdorff-lsc except the case where the range space is locally convex.

1. Lower semicontinuous multifunctions

The two most known concepts of lower semicontinuity for multifunctions are the lower semicontinuity in Vietoris sense (V-lsc) and the lower semicontinuity in Hausdorff sense (H-lsc). Given a set Y we denote by 2^{Y} the family of all subsets of Y. Every map F : X → 2^{Y} will be called a multifunction from X to Y. Now, let X and Y be two arbitrary topological spaces. We say that a multifunction F : X → 2^{Y} is V-lsc at a point x_{0} ∈ X provided for every open G ⊂ Y such that F (x_{0})∩G ≠ ∅ there exists a neighbourhood U(x_{0}) of x_{0} such that F (x) ∩ G ≠ ∅ for every x ∈ U(x_{0}). This is the first concept of lower semicontinuity.

Let (Y, U) be a uniform space. Recall that every uniformity generate a topology, and a topological space is uniformizable provided it is a Tichonov space. For multifunctions from X to (Y, U) we may formulate the second concept of lower semicontinuity. Namely, a multifunction F : X → 2^{Y} is called H-lsc at x_{0} ∈ X if for every W ∈ U there exists a neighbourhood U(x_{0}) of x_{0} such that

F (x_{0}) ⊂ W (F (x)) for every x ∈ U(x_{0}),

2000 Mathematics Subject Classification. 54C60, 54E15, 46A16. Key words and phrases. Lower semicontinuous multifunctions (set-valued maps), carte-

sian product, intersection, vector sum, and convex hull of multifunctions.

Andrzej Spakowski, On lower semicontinuous multifunctions in quasi-uniform and vector spaces, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 309–319, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204142.

where

W (F (x)) = { y ∈ Y : (z, y) ∈ W for some z ∈ F (x)}.

This is the second concept of lower semicontinuity. In particular, if Y is a topological vector space, with its natural uniformity generated by the neighbourhoods of 0, the condition of H-lsc can be written in the equivalent form:

F (x_{0}) ⊂ F (x) + V for every x ∈ U(x_{0}),

where V is a neighbourhood of 0 and C + D = {c + d : c ∈ C, d ∈ D} is the vector sum of sets C and D.

It is known that every topological space Y is quasi-uniformizable ([11, 9]). This means that there is family U of subsets of Y × Y such that:

(1) every U ∈ U contains the diagonal Δ of Y × Y,

(2) U, V ∈ U implies that U ∩ V ∈ U,

(3) for every U ∈ U there exists V ∈ U such that V ∘ V ⊂ U, where V ∘ V = {(x, y) ∈ Y × Y : (x, z), (z, y) ∈ V for some z ∈ Y },

(4) U ∈ U and U ⊂ V implies V ∈ U,

(5) the family {W (y) : W ∈ U, y ∈ Y } is a neighbourhood system

generating the topology of Y, where W (y) = {z ∈ Y : (y, z) ∈ W }. Every such family is called a quasi-uniformity of the topological space Y.

If Y is only a set, then a family U of subsets of Y ×Y satisfying properties (1)–(4) is called a quasi-uniformity on Y, and the pair (Y, U) a quasi-uniform space. A quasi-uniform space (Y, U) is a uniform space provided U have the following symmetric property: W ∈ U implies W^{−1} ∈ U, where W^{−1} = {(z, y) ∈ Y × Y : (y, z) ∈ W }.

In every topological space Y we have: for every A ⊂ Y and every quasiuniformity U of Y, cl (A) = ⋂{W^{−1}(A) : W ∈ U}. where cl (A) denotes the closure of A in Y. The usage of W^{−1} is explained by the following facts:

(1) the sets W (y), where W ∈ U and y ∈ Y, form a neighbourhood system for the topology of Y,

(2) for arbitrary A ⊂ Y we have: y ∈ cl (A), the closure of A, if and only if for every W ∈ U, W (y) ∩ A ≠ ∅ if and only if there exists a ∈ A such that a ∈ W (y), or equivalently y ∈ W^{−1}(a).

For a quasi-uniform (Y, U) space the definition of H-lsc should be modified as follows. A multifunction F : X → 2^{Y} is called H-lsc at x_{0} ∈ X if for every W ∈ U there exists a neighbourhood V of x_{0} such that

F (x_{0}) ⊂ W^{−1}(F (x)) for every x ∈ V.

Note that we have the following property: F is H-lsc at x_{0} if and only if cl (F) is H-lsc, where cl (F) is the closure multifunction of F, i.e. cl (F)(x) = cl (F (x)) for all x ∈ X. The basic relationships between V-lsc and H-lsc are well-known (see [4], [5]). Namely, if Y is a topological space, U a quasiuniformity of Y and F : X → 2^{Y} a multifunction, then H-lsc of F at x_{0} ∈ X implies its V-lsc at x_{0}. The converse holds provided the set F (x_{0}) is totally bounded. Recall that A ⊂ Y is called totally bounded provided for every W ∈ U there exists a finite set B ⊂ A such that A ⊂ W^{−1}(B). In general, V-lsc need not imply H-lsc.

2. B-lower semicontinuity

Penot [10] introduced a concept of bounded lower semicontinuity for multifunction from a topological space to a normed space. In [2] a similar idea is applied to convergence of sets, in particular, to convergence of epigraphs, with respect to the families of single subsets, compact subsets, weakly compact subsets, and of bounded subsets. Following this, we define an abstract concept of lower semicontinuity to unify the description of the above mentioned semicontinuities.

Let X be a topological space, (Y, U) a quasi-uniform space, B a cover of Y, i.e., a family of nonempty subsets of Y such that ⋃ B = Y. We say that F : X → 2^{Y} is B-lsc at x_{0} ∈ X provided for every W ∈ U and every B ∈ B there exists a neighbourhood U(x_{0}) of x_{0} such that

F (x_{0}) ∩ B ⊂ W^{−1}(F (x)) for every x ∈ U(x_{0}).

If F (x_{0}) = ∅, then F is trivially B-lsc at x_{0} for arbitrary cover B. Note also the following simple observations and remarks:

(1) If B = {Y }, then B-lsc is simply the H-lsc.

(2) If B is a cover of Y and B_{1} the family of all finite unions of subsets of B, then B-lsc implies the B_{1}-lsc.

(3) If B ⊂ B_{1}, then B_{1}-lsc implies B-lsc.

(4) If B is the family of all balls B(r) = {y ∈ Y : ||y|| < r} of a normed space Y, then B-lsc will be called also bounded H-lsc. This case is identical with the Penot’s concept [10].

(5) If for every B ∈ B the multifunction F_{B}(x) = F (x)∩B, x ∈ X, is Hlsc at x_{0}, then it is B-lsc at this point. The converse, in general, does not hold. For instance, we can take: F (0) = [0, 1] and F (x) = [0, 1) for x ∈ (0, 1], and B the family of all singletons of [0, 1]. Then F is B-lsc but F_{B}, where B = {1} is not H-lsc for F (x) ∩ {1} = ∅ for all x ∈ (0, 1]. For some positive results see [10].

Now, we show that if B is the family of all singletons of Y, or equivalently, the family of all finite subsets of Y, then B-lsc is simply the V-lsc.

Theorem 2.1. Let B be a cover of Y, F : X → 2^{Y}, and consider the following three statements:

(1) F is H-lsc at x_{0},

(2) F is B-lsc at x_{0},

(3) F is V-lsc at x_{0}. Then (1) ⇒ (2) ⇒ (3), and the converse implications does not hold.

Proof. That (1) implies (2) is clear because

F (x_{0}) ∩ B ⊂ F (x_{0}) ⊂ W^{−1}(F (x)).

Now assume that F is B-lsc at x_{0}. In virtue of B-lsc, we may assume that B contains all singletons of Y. We show that F is V-lsc at x_{0}. Given an open G ⊂ Y and y_{0} ∈ F (x_{0}) ∩ G we take W ∈ U such that W (y_{0}) ⊂ G. By the B-lsc there exists a neighbourhood U(x_{0}) of x_{0} such that

F (x_{0}) ∩ {y_{0}} ⊂ W^{−1}(F (x)) for all x ∈ U(x_{0}).

This implies that y_{0} ∈ W^{−1}(y) for some y ∈ F (x), or equivalently y ∈ W (y_{0}) for some y ∈ F (x). Consequently, F (x) ∩ G ≠ ∅ for all x ∈ U(x_{0}). This proves that F is V-lsc at x_{0}. It remains to show that the converse implications need not to hold. Let Y = ℝ^{2}, B the family of all straight lines of Y through the origin, and consider the following two multifunctions: F_{1}(t) the line {y = tx}, F_{2}(t) the line {y = 1 + tx}, for t ≥ 1. Observe that: F_{1} is V-lsc but not B-lsc at each point, while F_{2} is B-lsc but not H-lsc at each point. □

Theorem 2.2. Let F : X → 2^{Y} be a multifunction and B be the family of all singletons of Y. Then F is V-lsc at x_{0} if and only if it is B-lsc at x_{0}.

Proof. In virtue of the above theorem the second implication is clear. Now, assume that F is V-lsc at x_{0} and B is the family of all singletons of Y. We show that F is B-lsc at x_{0}. Let B = {y_{0}}. Then F (x_{0})∩B is empty or equal B. Let W ∈ U be arbitrary. Then W (y_{0}) is open and W (y_{0}) ∩ F (x_{0}) ≠ ∅. By the V-lsc of F at x_{0} there exists a neighbourhood U(x_{0}) of x_{0} such that W (y_{0}) ∩ F (x) ≠ ∅ for all x ∈ U(x_{0}). Thus for every x ∈ U(x_{0}) there exists y ∈ W (y_{0}) ∩ F (x). From this we infer that (y_{0}, y) ∈ W, or equivalently (y, y_{0}) ∈ W^{−1}. Consequently, y_{0} ∈ W^{−1}(y) ⊂ W^{−1}(F (x)). This proves that F is B-lsc at x_{0}. □

A general, similar theorem exists for B-lsc and V-lsc. First, let us introduce a generalized concept of a totally bounded set. Let B be a cover of Y, and (Y, U) a quasi-uniform space. A set A ⊂ Y will be called B-totally bounded if for every B ∈ B, the set A ∩ B is totally bounded. Note that, if B = {Y }, then the B-total boundedness is simply the usual total boundedness, and if B is the family of all singletons, then the B-total boundedness is trivial: each subset of Y is B-totally bounded.

Theorem 2.3. Let F : X → 2^{Y}, x_{0} ∈ X, and F (x_{0}) be B-totally bounded, where B is a cover of Y. Then F is V-lsc at x_{0} if and only if it is B-lsc at x_{0}.

Proof. It is clear that B-lsc implies V-lsc. Now, let us assume that F is V-lsc at x_{0} and F (x_{0}) is B-totally bounded. Let B ∈ B and W ∈ U be arbitrary. Let V ∈ U be such that V ∘ V ⊂ W. Hence V^{−1} ∘ V^{−1} ⊂ W^{−1}, and there exist y_{1}, …, y_{n} ∈ F (x_{0}) such that

F (x_{0}) ∩ B ⊂ ⋃_{i=1}^{n} V^{−1}(y_{i}) = ⋃_{i=1}^{n} V^{−1}(F (x_{0}) ∩ {y_{i}}).

In virtue of the V-lsc at x_{0} there exists a neighbourhood U(x_{0}) of x_{0} such that F (x_{0}) ∩ {y_{i}} ⊂ V^{−1}(F (x)), for i = 1, …, n, and x ∈ U(x_{0}). This implies that F (x_{0}) ∩ B ⊂ V^{−1}(V^{−1}(F (x)) ⊂ W^{−1}(F (x)), which ends the proof. □

As a corollary, for B = {Y }, we get the well-known equivalence between V-lsc and H-lsc whenever we consider totally-bounded valued multifunctions.

Remarks.

One can examine a concept of B-usc described as follows: for every B ∈ B and every W ∈ U there exists a neighbourhood U(x_{0}) of x_{0} such that F (x) ∩ B ⊂ W (F (x_{0})) for every x ∈ U(x_{0}). It is clear that V-usc always implies B-usc. Unfortunately, if B is the family of all singletons, B-usc multifunctions need not to be V-usc. For example, let Y = ℝ^{2}, B the family of all singletons of Y, and we take the following multifunction: F (0) is the line {y = 0}, and F (t) is the line {y = tx}, for t > 0. F is B-usc at 0 but not V-usc at this point.

See [3] for the bounded H-usc, i.e. for the B-usc with B being the family of all bounded subsets of Y.

It is clear that B-lsc is topologizable provided B is the family of all singletons or B = {Y }. In general, B-lsc is not topologizable. To show this we use the following Diagonalization Criterion (see e.g. [4]):

Theorem 2.4. Let T be a directed set and for each t ∈ T there is another directed set E(t). Then we define a new directed set D = T × Π_{t∈T} E(t) ordered as follows: (t, (α(t))) ≤ (s, (β(t))) if and only if t ≤ s and α(t) ≤ β(t) for each t ∈ T. Suppose that z(t, γ), t ∈ T, γ ∈ E(t), are elements of a topological space Z. Consider the following net: z(t, α) = z(t, α(t)), (t, α) ∈ D, where α(t) is the t-coordinate of α. If lim_{t∈T} lim_{γ∈E(t)} z(t, γ) = z then lim_{(t,α)∈D} z(t, α(t)) = z.

Now, we can construct an example of a non-topologizable B-lsc. Observe first that B-lsc is simply continuity with respect to the following B^{−}- convergence: A_{λ} → A_{0} whenever for every W ∈ U and every B ∈ B there exists λ_{0} such that

A_{0} ∩ B ⊂ W^{−1}(A_{λ}) for every λ > λ_{0}.

Let T = ℕ and for each n ∈ T we take E(n) = ℕ. Let Y = ℝ^{2} and B be the family of all straight line through the (0, 0). For every n, k ∈ ℕ we denote: A(n, k) = the line {y = (1/k)x + 1/n}, A(n) = the line {y = 1/n}, A(0) = B(0) = the line {y = 0}. It is easy to check B^{−}-convergence: lim_{n} lim_{k} A(n, k) = lim_{n} A(n) = A(0).

On the other hand the convergence lim_{(n,α)} A(n, α(n)) = A_{0} does not holds for A(0)∩B(0) = the line {y = 0} is not contained in any A(n, α(n))+ V, where V is a neighbourhood of (0, 0). Thus the B^{−}-convergence is not topologizable.

3. Unions and cartesian products

In this paragraph we deal with some set-theoretical operations on multifunctions, namely with unions and cartesian products (see [4]). Operation of intersection of multifunctions will be examined separately in the next paragraph.

Unions. Let X and Y be spaces and F_{i} : X → 2^{Y}, i ∈ I, a family of multifunctions. The union F = ⋃_{i∈I} F_{i} of multifunctions F_{i} is defined by F (x) = ⋃_{i∈I} F_{i}(x), x ∈ X. It is known and easy to prove that the union of an arbitrary family of V-lsc multifunctions is V-lsc. However, the union of an infinite family of H-lsc multifunctions need not to be H-lsc. For instance, define F_{n}(0) = [0, n] and F_{n}(x) = [0, 1/x] for x ∈ (0, 1], n = 1, 2, . . .. Then the multifunctions F_{n} are H-lsc at 0. But, their union F = ⋃_{n} F_{n} is not H-lsc at 0 for F (0) = [0, +∞).

Theorem 3.1. Let X be a topological space, (Y, U) a quasi-uniform space and F_{1}, F_{2} : X → 2^{Y} multifunctions B-lsc at x_{0} ∈ X. Then the union multifunction F = F_{1} ∪ F_{2} is B-lsc at x_{0}.

Proof. The proof is a consequence of the formula:

W^{−1}(F_{1}(x)) ∪ W^{−1}(F_{2}(x)) = W^{−1}(F_{1}(x) ∪ F_{2}(x)),

where W ∈ U and x ∈ X. □

It is clear that the above theorem holds for finitely many multifunctions, and need not hold when we consider an infinite family of multifunctions.

Products. Now, let us describe the cartesian product of multifunctions. Let X and Y_{1}, Y_{2} be spaces and F_{i} : X → 2^{Y_{i}}, i = 1, 2, multifunctions. The product of two multifunctions F_{1} and F_{2} is defined as the multifunction F = F_{1} × F_{2} : X → 2^{Y_{1}×Y_{2}} such that F (x) = F_{1}(x) × F_{2}(x), x ∈ X. In particular, if F_{2}(x) = Y_{2} for all x, or F_{1}(x) = Y_{1} for all x, we will write simply, F_{1} ×Y_{2}, or Y_{1} ×F_{1}, respectively. Analogously, we define the product Π_{i∈I}F_{i} of an arbitrary family of multifunctions F_{i}, i ∈ I. It is known that the product of an arbitrary family of V-lsc (H-lsc) multifunctions is also V-lsc (H-lsc). Remark that the product of H-lsc multifunctions has more complicated nature than the product of V-lsc ones. To formulate a general theorem for B-lsc we need to consider the product of quasi-uniform spaces. First, we describe the product of two quasi-uniform spaces. Let (Y_{i}, U_{i}), i = 1, 2, be quasi-uniform spaces, and P_{i} : Y_{1} × Y_{2} → Y_{i}, i = 1, 2, be the projections, i.e. P_{i}(y_{1}, y_{2}) = y_{i}, i = 1, 2. By the product quasi-uniformity U = U_{1} × U_{2} in Y = Y_{1} × Y_{2} we mean a quasi-uniformity generated by the base consisting of sets of the form

[W_{1}, W_{2}] = {(z_{1}, z_{2}) ∈ Y × Y : (P_{i}(z_{1}), P_{i}(z_{2})) ∈ W_{i}, i = 1, 2},

where W_{i} ∈ U_{i}, i = 1, 2. In other words, the set [W_{1}, W_{2}] has a form:

{(s_{1}, y_{1}, s_{2}, y_{2}) ∈ Y_{1} × Y_{2} × Y_{1} × Y_{2} : (s_{1}, s_{2}) ∈ W_{1}, (y_{1}, y_{2}) ∈ W_{2}}.

Remark that [W_{1}, W_{2}] = W̃_{1} ∩ W̃_{2}, where

W̃_{i} = {(z_{1}, z_{2}) ∈ Y × Y : (P_{i}(z_{1}), P_{i}(z_{2})) ∈ W_{i}}, i = 1, 2.

The sets W̃_{i}, W_{i} ∈ U_{i}, i = 1, 2, form a subbase of the product quasiuniformity U_{1}×U_{2}. In case of an arbitrary family of quasi-uniform spaces we proceed similarly as above and as in the construction of product topological structures. Let (Y_{i}, U_{i}), i ∈ I, be a family od quasi-uniform spaces. Denote: Y = Π_{i∈I}Y_{i}, P_{i} the projection on the i-th axis, i.e. P_{i}(y) = y_{i}, where y_{i} is the i-th coordinate of y ∈ Y, i ∈ I. By the product quasi-uniformity U = Π_{i∈I}U_{i} in Y we mean a quasi-uniformity generated by the subbase consisting of sets of the form

W̃_{i} = {(z_{1}, z_{2}) ∈ Y × Y : (P_{i}(z_{1}), P_{i}(z_{2})) ∈ W_{i}},

where W_{i} ∈ U_{i}, i ∈ I. Observe that if B_{i} is a cover of Y_{i}, i ∈ I, then Π_{i∈I}B_{i}, i.e., the family of all sets of the form Π_{i∈I}B_{i} with B_{i} ∈ B_{i} and B_{i} = Y_{i} for all but a finite number of i ∈ I, is a cover of Π_{i∈I}Y_{i}.

Theorem 3.2. Let (Y_{i}, U_{i}), i = 1, 2, be quasi-uniform spaces and A_{i} ⊂ Y_{i}, i = 1, 2, be arbitrary subsets. Then

(1) [W_{1}, W_{2}](A_{1} × A_{2}) = W_{1}(A_{1}) × W_{2}(A_{2}),

(2) [W_{1}, W_{2}]^{−1}(A_{1} × A_{2}) = W_{1}^{−1}(A_{1}) × W_{2}^{−1}(A_{2}).

Proof. We have [W_{1}, W_{2}](A_{1} × A_{2})

= {(s_{1}, s_{2}) ∈ Y_{1} × Y_{2} : ∃ y_{1} ∈ A_{1} ∃ y_{2} ∈ A_{2} (y_{1}, y_{2}, s_{1}, s_{2}) ∈ [W_{1}, W_{2}]} = {(s_{1}, s_{2}) ∈ Y_{1} × Y_{2} : ∃ y_{1} ∈ A_{1} ∃ y_{2} ∈ A_{2} (y_{1}, s_{1}) ∈ W_{1}, (y_{2}, s_{2}) ∈ W_{2}} = {(s_{1}, s_{2}) ∈ Y_{1} × Y_{2} : s_{1} ∈ W_{1}(A_{1}), s_{2} ∈ W_{2}(A_{2})} = W_{1}(A_{1}) × W_{2}(A_{2}),

which proves (1). The proof of (2) is similar. □

Theorem 3.3. Let X be a topological space, (Y_{i}, U_{i}), i ∈ I, a family of quasi-uniform spaces and F_{i} : X → 2^{Y_{i}} a multifunction B_{i}-lsc at x_{0} ∈ X, where B_{i} is a cover of Y_{i}, i ∈ I. Then the product multifunction Π_{i∈I}F_{i} is Π_{i∈I}B_{i}-lsc at x_{0}.

Proof. By the construction of the product of quasi-uniform spaces (Y_{i}, U_{i}), i ∈ I, it is sufficient to consider only the case I = {1, 2}. In general case, the proof is similar. Let multifunctions F_{i} : X → 2^{Y_{i}}, i = 1, 2, be B_{i}-lsc at x_{0} ∈ X and W ∈ U_{1} × U_{2} be arbitrary. There exist W_{i} ∈ U_{i}, i = 1, 2, such that [W_{1}, W_{2}] ⊂ W. Now, let B_{i} ∈ B_{1}. There exists a neighbourhood U(x_{0}) of x_{0} such that F_{i}(x_{0}) ∩ B_{i} ⊂ W_{i}^{−1}(F_{i}(x)), i = 1, 2, x ∈ U(x_{0}), and, by the Lemma 3.2, we get

(F_{1}(x_{0}) ∩ B_{1}) × (F_{2}(x_{0}) ∩ B_{2}) ⊂ [W_{1}, W_{2}]^{−1}(F_{1}(x) × F_{2}(x)),

for every x ∈ U(x_{0}). This shows the B_{1} × B_{2}-lsc of F_{1} × F_{2} because

(F_{1}(x_{0}) ∩ B_{1}) × (F_{2}(x_{0}) ∩ B_{2}) = (F_{1}(x_{0}) × F_{2}(x_{0})) ∩ (B_{1} × B_{2}).
□

Remark. The converse theorem also holds. Namely, if a product multifunction Π_{i∈I}F_{i} is Π_{i∈I}B_{i}-lsc at x_{0}, then for every i ∈ I the multifunction F_{i} is B_{i}-lsc at x_{0}.

4. Intersections

In optimization theory the lower semicontinuity properties of intersections of multifunctions play an important role [10]. The most wanted theorems are ones with no boundedness conditions on the values of intersecting multifunctions. Here we formulate a theorem of this kind. Let Y be a normed space. If we assume that the considered multifunctions are boundedly Hlsc, i.e., B-lsc with B being the family of all balls B(r), r > 0, then we may formulate a theorem on intersection, without boundedness conditions on F (x_{0}). For this we need a lemma from [6] on interiority properties of convex, bounded, and with the nonempty interior subsets of a normed space.

Theorem 4.1. Let Y be a normed space and A ⊂ Y be convex, bounded, and with the nonempty interior. Then for every ε > 0 there exist a set C ⊂ int(A) and δ > 0 such that C + B(δ) ⊂ A ⊂ C + B(ε).

We need also the following well-known (see [12, 14]) and very useful law of cancellation:

Theorem 4.2. Let A, B and C be subsets of a topological vector space Y. Assume that B is bounded, and C is nonempty, closed and convex. Then A + B ⊂ cl (C + B) implies A ⊂ C. In particular, A + B ⊂ C + B implies A ⊂ C, and cl (A + B) ⊂ cl (C + B) implies A ⊂ C.

Let X be a topological space and Y a topological vector space. A multifunction F from X to Y is called locally convex-valued (locally closed-valued) at x_{0} ∈ X if there is a neighbourhood U of x_{0} such that F (x) is convex (closed) for every x ∈ U.

Theorem 4.3. Let X be a topological space, Y a normed space, B the family of balls B(r) ⊂ Y, 0 < r, F_{1} and F_{2} two multifunctions from X to Y and F = F_{1} ∩ F_{2}. If F_{1} and F_{2} are B-lsc at x_{0} ∈ X, locally convex- and locally closed-valued at x_{0}, and int F (x_{0}) ≠ ∅, then F is B-lsc at x_{0}, and hence, V-lsc at x_{0}.

Proof. By the assumption on the interior of F (x_{0}) in Y there exists r > 0 such that

(1)  int (F (x_{0}) ∩ B(r)) ≠ ∅.

Let ε > 0 be arbitrary. By the Lemma 4.1 there exist a subset C ⊂ F (x_{0}) ∩ B(r) and δ > 0 such that C + B(δ) ⊂ F (x_{0}) ∩ B(r) ⊂ C + B(ε). In virtue of the B-lsc at x_{0} there exists a neighbourhood U(x_{0}) of x_{0} such that

C + B(δ) ⊂ F_{i}(x_{0}) ∩ B(r) ⊂ F_{i}(x) + B(δ) for x ∈ U(x_{0}) and i = 1, 2.

We can assume that the multifunctions are closed- and convex-valued on U(x_{0}). Applying the law of cancellation, we infer that C ⊂ F_{1}(x) ∩ F_{2}(x) for every x ∈ U(x_{0}). But this implies that

\begin{aligned}F_{1}(x_{0}) ∩ F_{2}(x_{0}) ∩ B(r) & ⊂ & C + B(ε) \\ & ⊂ & F_{1}(x) ∩ F_{2}(x) + B(ε) \end{aligned}

for all x ∈ U(x_{0}). This shows that the intersection F = F_{1} ∩ F_{2} is B-lsc at x_{0} and ends the proof. □

Remark. If Y is finite dimensional we can omit in the above theorem the assumption that the multifunctions are locally closed-valued, and then proceed in a manner as in [6] using the below theorem on local interior property of B-lower semicontinuous multifunctions.

5. Vector operations

Here we consider vector sum and convex hull operations on lower semicontinuous multifunctions with values in a topological vector space (see e.g. [7]). Let X be a topological space, Y a topological vector space and F, G : X → 2^{Y}. We define two multifunctions:

\begin{aligned}(F + G)(x) & = & F (x) + G(x) \\ & = & \{a + b : a ∈ F (x), b ∈ G(x)\}, x ∈ X, \end{aligned}

called the vector sum of F and G, and conv (F)(x) = conv (F (x)) the convex hull of F (x), x ∈ X, called the convex hull of F.

Vector sum. It is known and easy to prove that the vector sum of two H-lsc multifunctions is H-lsc. We state some further results and show that, in general, the vector sum of two B-lsc multifunctions need not to be B-lsc.

Theorem 5.1. Let F, G : X → 2^{Y} be V-lsc at x_{0} ∈ X. Then the vector sum F + G is V-lsc at x_{0}.

Proof. Let V be an arbitrary neighbourhood of 0 in Y, F and G be V-lsc at x_{0}, and recall that V-lsc is equivalent to B-lsc with B equals the family of all singletons of Y. Let b ∈ Y be such that (F (x_{0}) + G(x_{0})) ∩ {b} ≠ ∅. Then b = b_{1} + b_{2} with b_{1} ∈ F (x_{0}) and b_{2} ∈ G(x_{0}). By the B-lsc of F and G there exists a neighbourhood U(x_{0}) such that

\begin{matrix}F (x_{0}) ∩ \{b_{1}\} ⊂ F (x) + V and \\ G(x_{0}) ∩ \{b_{2}\} ⊂ G(x) + V \end{matrix}

for all x ∈ U(x_{0}). This implies that

b_{1} + b_{2} ∈ F (x) + G(x) + V + V

for x ∈ U(x_{0}), or equivalently,

(F (x_{0}) + G(x_{0})) ∩ {b} ⊂ F (x) + G(x) + V + V

for x ∈ U(x_{0}), which shows B-lsc of F + G at x_{0} and ends the proof. □

We say that a cover B of Y is translation invariant if B + c ∈ B for every B ∈ B and every vector c ∈ Y.

Theorem 5.2. Let B be a translation invariant cover of Y, F : X → 2^{Y} a multifunction B-lsc at x_{0} ∈ X and g : X → Y a function continuous at x_{0}. Then the vector sum (F + g)(x) = F (x) + g(x), x ∈ X, is a multifunction B-lsc at x_{0}.

Proof. Let V be a neighbourhood of 0 in Y and B ∈ B. Note that

(F (x_{0}) + g(x_{0})) ∩ B = F (x_{0}) ∩ (B − g(x_{0})) + g(x_{0})

By the assumptions for all x in a neighbourhood of x_{0} we have

\begin{aligned}(F (x_{0}) + g(x_{0})) ∩ B & ⊂ & (F (x) + g(x_{0}) + V) ∩ B \\ & ⊂ & (F (x) + g(x) + V + V). \end{aligned}

This shows B-lsc of F + g at x_{0} and ends the proof. □

The following example shows that translation invariantness of B is not sufficient to get B-lsc of the vector sum of a two B-lsc multifunctions.

Example. Let Y = ℝ^{3}, B consists only of the plane {y = x} and all of its translations. Define two multifunctions: F (t) the line {z = ty, x = 0} and G(t) the line {y = 0, z = 0}, t ≥ 0. Observe that F and G are B-lsc but their vector sum F + G is not B-lsc at each point.

Convex hull. It is known [8] and easy to proof that for every V-lsc multifunction F, the convex hull of F is also V-lsc. We use the concept of B-lsc to get a general result provided the space Y is locally convex. In particular, we get a result for H-lsc.

Theorem 5.3. Let Y be a locally convex space and F : X → 2^{Y} be B-lsc at x_{0} ∈ X. Then the convex hull of F is B-lsc at x_{0}.

Proof. Let V ⊂ Y be a convex neighbourhood of 0 in Y and B ∈ B. By the B-lsc of F at x_{0} there exists a neighbourhood U(x_{0}) of x_{0} such that

(2)  F (x_{0}) ∩ B ⊂ F (x) + V for all x ∈ U(x_{0}).

We claim that

conv (F (x_{0})) ∩ B ⊂ conv (F (x)) + V for all x ∈ U(x_{0}).

Indeed, let y ∈ conv (F (x_{0}))∩B and x ∈ U(x_{0}) be arbitrary. In virtue of (2) there exist n ∈ ℕ, y_{1}, …, y_{n} ∈ F (x_{0}), z_{1}, …, z_{n} ∈ F (x), v_{1}, …, v_{n} ∈ V, and positive numbers t_{1}, …, t_{n} such that y = t_{1}y_{1} + ⋯ + t_{n}y_{n}, t_{1} + ⋯ + t_{n} = 1, and y_{i} = z_{i} + v_{i} for every 1 ≤ i ≤ n. This implies that

y = t_{1}z_{1} + ⋯ + t_{n}z_{n} + t_{1}v_{1} + ⋯ + t_{n}v_{n} ∈ conv (F (x)) + V

for V is convex, which ends the proof. □

Remark. If the topological vector space Y is not locally convex then the convex hull operation does not preserve H-lsc. Indeed, if Y is metrizable and not locally convex then there exists a sequence y_{n} ∈ Y which converges to 0 such that the convex hull of the set {y_{n} : n = 1, 2, …} is not bounded [1]. Now, observe that the multifunction F defined by: F (0) = {0, y_{1}, y_{2}, …}, F (1/n) = {0, y_{1}, …, y_{n}}, n = 1, 2, …, is H-lsc at 0 but the convex hull of F is not. For a simple example of such sequence y_{n} ∈ l^{p} (0 < p < 1) see [13] or [1].

References

[1] Tetsu Araki, A characterization of non-local convexity in some class of topological vector spaces, Math. Japon. 41 (1995), no. 3, 573–577. MR 96f:46003

[2] Jonathan M. Borwein and Jon D. Vanderwerff, Epigraphical and uniform convergence of convex functions, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1617–1631. MR 96j:49012

[3] C. Costantini, A splitting property of the upper bounded-Hausdorff convergence, Set-Valued Anal. 2 (1994), no. 1-2, 135–139. MR 95e:54013

[4] Erwin Klein and Anthony C. Thompson, Theory of correspondences, John Wiley & Sons Inc., New York, 1984. MR 86a:90012

[5] Alojzy Lechicki, On continuous and measurable multifunctions, Comment. Math. Prace Mat. 21 (1979), no. 1, 141–156. MR 81j:54025

[6] Alojzy Lechicki and Andrzej Spakowski, A note on intersection of lower semicontinuous multifunctions, Proc. Amer. Math. Soc. 95 (1985), no. 1, 119–122. MR 87d:54033

[7] Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 13,54f

[8] , Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 17,990e [9] M. G. Murdeshwar and S. A. Naimpally, Quasi-uniform topological spaces, P. Noord-

hoff Ltd., Groningen, 1966. MR 35 #2267

[10] J.-P. Penot, Preservation of persistence and stability under intersections and operations. I. Persistence, J. Optim. Theory Appl. 79 (1993), no. 3, 525–550. MR 94k:49013

[11] William J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316–317. MR 25 #3506b

[12] M. G. Rabinovich, Some classes of spaces of convex sets and their extensions, Sib. Math. Zh. 8 (1967), 1405–1415, English translation in Siberian. Math. J. 8 (1967) 1064–1070.

[13] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York, 1973. MR 51 #1315

[14] R. Urbański, A generalization of the Minkowski-Rȧdström-Hörmander theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 9, 709–715. MR 56 #1027

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