Points of upper and lower semicontinuity of multivalued functions ..................
We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely. among other results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping $F : X \times Y \rightarrow \scr K (Z)$ is jointly upper semicontinuous o...
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Institute of Mathematics, NAS of Ukraine
2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507630045233152 |
|---|---|
| author | Mirmostafaee, A. K. Мірмостафае, А. К. |
| author_facet | Mirmostafaee, A. K. Мірмостафае, А. К. |
| author_sort | Mirmostafaee, A. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:26:20Z |
| description | We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely. among other results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping $F : X \times Y \rightarrow \scr K (Z)$ is jointly upper semicontinuous on sets of the from $D \times \{ y_0\}$, where $D$ is a dense G\delta subset of $X$ and $y_0 \in Y$.
A similar result is obtained for the joint lower semicontinuity of upper horizontally quasicontinuous mappings. These
results improve some known results on the joint continuity of single-valued functions. |
| first_indexed | 2026-03-24T02:12:22Z |
| format | Article |
| fulltext |
UDC 515.12
A. K. Mirmostafaee (Ferdowsi Univ. Mashhad, Iran)
POINTS OF UPPER AND LOWER SEMICONTINUITY
OF MULTIVALUED FUNCTIONS*
ТОЧКИ ВЕРХНЬОЇ ТА НИЖНЬОЇ НАПIВНЕПЕРЕРВНОСТI
ДЛЯ БАГАТОЗНАЧНИХ ФУНКЦIЙ
We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely. among other
results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping F : X \times Y \rightarrow
\rightarrow \scrK (Z) is jointly upper semicontinuous on sets of the from D\times \{ y0\} , where D is a dense G\delta subset of X and y0 \in Y.
A similar result is obtained for the joint lower semicontinuity of upper horizontally quasicontinuous mappings. These
results improve some known results on the joint continuity of single-valued functions.
Вивчається спiльна верхня та нижня напiвнеперервнiсть для багатозначних функцiй двох змiнних. Бiльш точно,
серед iнших результатiв показано, що за деяких умов нижньо горизонтально квазiнеперервне вiдображення вiд двох
змiнних F : X \times Y \rightarrow \scrK (Z) є спiльно верхньо напiвнеперервним на множинах з D \times \{ y0\} , де D — щiльна G\delta
пiдмножина X та y0 \in Y. Подiбний результат отримано також для спiльної нижньої напiвнеперервностi верхньо
горизонтальних квазiнеперервних вiдображень. Цi результати покращують деякi вiдомi результати про спiльну
неперервнiсть однозначних функцiй.
1. Introduction and preliminaries. Throughout the paper, we will assume that all topological spaces
are T1. For a topological space Z, we denote by \scrP (Z), \scrC (Z) and \scrK (Z) the set of all nonempty
subsets, the set of all nonempty closed subsets and the set of all nonempty compact subsets of Z
respectively. If F : X \rightarrow \scrP (Z) is a set-valued function. For a subset G of Z, we define F+(G) and
F - (G) as follows:
F+(G) = \{ x \in X : F (x) \subseteq G\} , F - (G) = \{ x \in X : F (x) \cap G \not = \varnothing \} .
The function F is called:
(a) upper (resp. lower) semicontinuous if for every open subset G of Z, F+(G) (resp. F - (G))
is an open subset of X.
(b) upper (resp. lower) quasicontinuous at x0 \in X if for any open set G, with x0 \in F+(G)
(resp. x0 \in F - (G)) and any neighborhood U of x0, there exists a nonempty open set V \subseteq U such
that V \subseteq F+(G) (resp. V \subseteq F - (G)).
(c) categorically upper (resp. lower) quasicontinuous at x0 \in X, if for each neighborhood U of
x0 and neighborhood G containing F (x0), there exists a set A \subseteq U of the second category such that
F (a) \subseteq G (resp. F (a) \cap G \not = \varnothing ) for all a \in A.
Let Z be a topological space and \{ \scrG n\} be a sequence of open covers of Z. For every z \in Z and
n \in \BbbN , let St(z,\scrG n) = \cup \{ G \in \scrG n : z \in \scrG n\} . The sequence \{ \scrG n\} is called:
(a) A development, if for every z \in Z, the sequence \{ St(z,\scrG n)\} is a base at z. A space with
a developable space is called a developable space. A regular developable space is called a Moore
space.
* This paper was supported by Ferdowsi University of Mashhad (Grant No. 2/42133).
c\bigcirc A. K. MIRMOSTAFAEE, 2017
1224 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
POINTS OF UPPER AND LOWER SEMICONTINUITY OF MULTIVALUED FUNCTIONS 1225
(b) A weak k-development if for every compact subset K \subset Z and \{ \scrH n\} such that for every
n, \scrH n \subseteq \scrG n, \scrH n is finite, K \cap H \not = \varnothing for every H \in \scrH n and K \subseteq
\bigcup
\scrH n, we have that the
sequence
\bigl\{ \bigcap
i\leq n
\bigl( \bigcup
\scrH i
\bigr) \bigr\}
is a base at K. A space with a weak k-development is called a weakly
k-developable space.
We refer the reader to [1] for more information about k-developable topological spaces.
The aim of this paper is to obtain points of joint upper and lower semicontinuity of two variable
set-valued functions. One of the first results in this direction is the following.
Theorem 1.1 [4]. Let X be a Baire space, Y a second countable space, Z be a metrizable
space and F : X \times Y \rightarrow \scrK (Z) be a compact-valued mapping which is lower semicontinuous with
respect to the first variable and upper semicontinuous with respect to the second variable. Then there
exists a dense in X G\delta -set A \subseteq X such that F is jointly upper semicontinuous at every point of the
set A\times Y.
Let X and Y be topological spaces and let f : X \rightarrow Y be a function. The function f is called
quasicontinuous if for each x \in X and neighborhoods U of x and V of f(y), there is a nonempty
open subset U \prime of U such that f(U \prime ) \subseteq V.
The notion of quasicontinuity turned out to be a useful tool in some mathematical problems (see,
e.g., [2, 3, 10, 12, 13, 17, 18, 20, 21]).
In 1975, V. Popa [22] generalized the notion of quasicontinuity for set-valued functions. Since
then, some authors investigated various types of continuity of set-valued functions [5 – 9].
A function f : X \times Y \rightarrow Z is said to be horizontally quasicontinuous at (x, y) \in X \times Y if
for any neighborhoods U, V and W of x, y and f(x, y) respectively, there is a nonempty open
subset U \prime \subset of X and y\prime \in V such that f(U \prime \times \{ y\prime \} ) \subseteq W. In [14 – 16] some properties of
horizontally quasicontinuous functions are investigated. In Section 2, we will define upper and
lower horizontal quasicontinuity for two-variable set-valued functions. We will show that if F is
a lower horizontally quasicontinuous function, Fx : Y \rightarrow \scrK (Z) is upper semicontinuous and F y :
X \rightarrow \scrK (Z) is categorically upper quasicontinuous for each (x, y) \in X \times Y. Then there is a dense
G\delta subset D of X such that F is jointly upper semicontinuous on D \times \{ y0\} provided that X is
Baire, y0 \in Y has a countable base and Z is a normal weakly k-developable space.
Finally, in Section 3, we will investigate lower semicontinuity of a two variable set-valued
function F : X \times Y \rightarrow \scrC (Z). In fact, by using a different method from what we used in Section 2,
we will prove a similar result for upper horizontally quasicontinuous functions. In fact, we will
show that if F : X \times Y \rightarrow \scrC (Z) is an upper horizontally quasicontinuous function such that Fx :
Y \rightarrow \scrC (Z) is lower semicontinuous and F y : X \rightarrow \scrC (Z) is categorically lower quasicontinuous for
each (x, y) \in X \times Y. Then F is symmetrically lower quasicontinuous at each point of X \times \{ y0\}
provided that X is Baire, y0 \in Y has a countable base and Z is regular. Our results can be considered
as generalizations of corresponding results in [5, 11, 15, 16].
2. Upper semicontinuity of two variable set-valued functions. In this section, we will show
that under some circumstances, a two-variable function set-valued function F : X \times Y \rightarrow \scrK (Z), for
each y0 \in Y, is jointly upper-semicontinuous on a set of the form D\times \{ y0\} , where D is a dense G\delta
subset of X.
We begin with the following definitions.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1226 A. K. MIRMOSTAFAEE
Definition 2.1. Let F : X \times Y \rightarrow \scrP (Z) be a function and (x0, y0) \in X \times Y. The function F is
called:
(a) symmetrically upper (resp. lower) quasicontinuous at (x0, y0) with respect to the second
variable if for each nonempty open subset W of Z with (x0, y0) \in F+(W ) (resp. (x0, y0) \in F - (W ))
and a neighborhood U of x0, there are a nonempty open subset U \prime of U and a neighborhood V of
y0 such that U \prime \times V \subseteq F+(W ) (resp. U \prime \times V \subseteq F - (W ));
(b) upper (resp. lower) horizontally quasicontinuous at (x0, y0) if whenever U, V and W are
open subsets X, Y and Z respectively with (x0, y0) \in U\times V and (x0, y0) \in F+(W ) (resp. (x0, y0) \in
\in F - (W )), there are a nonempty open subset U \prime of U and y\prime \in V such that (x, y\prime ) \in F+(W )
(resp. (x, y\prime ) \in F - (W )) for each x \in U \prime .
Now, we are ready to state one of the main results of this section.
Theorem 2.1. Let X be a Baire space, Y a space and Z a space with a weakly k-development.
Let F : X \times Y \rightarrow \scrK (Z) be symmetrically upper quasicontinuous with respect to the second variable
at each point of X \times \{ y0\} for some y0 \in Y. Then there is a dense G\delta subset D of X such that F is
jointly upper semicontinuous at each point of D \times \{ y0\} .
Proof. Let \{ \scrG n : n \in \BbbN \} be a k-weak development for Z. For each natural number n, let Dn
be the set of all x \in X such that for some finite subset \scrH of \scrG n and neighborhoods U of x, V of
y0 respectively, we have U \times V \subseteq F+(
\bigcup
\scrH ) and F (t, y0) \cap H \not = \varnothing for each H \in \scrH and t \in U.
By the definition, each Dn is open in X. We will show that each Dn is dense in X. Let n be a
fixed natural number and U be a nonempty open subset of X.
We say that the property p(k), k \in \BbbN , holds if the following statement is true.
If there is a nonempty open subset U \prime of U such that F (U \prime , y0) \subseteq
\bigcup
\scrH , where \scrH \subseteq \scrG n with
| \scrH | \leq k, then Dn contains a nonempty open subset of U.
Let U \prime be a nonempty open subset of U such that F (U \prime , y0) \subseteq H for some H \in \scrG n. By upper
symmetrical quasicontinuity of F on U \prime \times \{ y0\} , we can find a nonempty open subset U \prime \prime of U \prime and
a neighborhood V of y0 such that F (x, y) \subseteq H for each (x, y) \in U \prime \prime \times V. Therefore U \prime \prime \subseteq Dn.
Thus p(1) is true.
Let p(k) hold for some k and U \prime be a nonempty open subset of U such that F (U \prime , y0) \subseteq
\bigcup
\scrH
for some subset \scrH of \scrG n with | \scrH | = k + 1. If for some x0 \in U \prime , there is a proper subset of \scrH ,
say \scrH \prime , such that F (x0, y0) \subseteq
\bigcup
\scrH \prime , then | \scrH \prime | \leq k and by symmetrically upper quasicontinuity of
F, we can find a nonempty open subset U \prime \prime of U \prime such that F (U \prime \prime , y0) \subseteq
\bigcup
\scrH \prime . Therefore, by our
hypothesis, Dn contains a nonempty open subset of U. So that we may assume that \scrH has exactly
k + 1 elements say H1, . . . ,Hk+1 and F (x, y0) \cap Hi \not = \varnothing for each x \in U \prime and 1 \leq i \leq k + 1. By
symmetrical upper quasicontinuity of F on U \prime \times \{ y0\} , we can find a nonempty open subset U \prime \prime of
U \prime and a neighborhoods V of y0 such that F (U \prime \prime , V ) \subseteq
\bigcup
\scrH . Then
F (U \prime \prime \times V ) \subseteq
\bigcup
\scrH , F (x, y0) \cap Hi \not = \varnothing \forall x \in U \prime \prime .
Therefore U \prime \prime \subseteq Dn. Hence p(k) holds for all k \in \BbbN .
Now, if x \in U and F (x, y0) \subseteq
\bigcup
\scrH , where \scrH is a finite subset of \scrG n. By symmetrically upper
quasicontinuity of F at (x, y0), there is a nonempty open subset U \prime of U and a neighborhood V
of y0 such that F (U \prime \times V ) \subseteq
\bigcup
\scrH . According to p(k), where k = | \scrH | , the set Dn contains a
nonempty subset of U. This shows that Dn is dense in X. Let D =
\bigcap \infty
n=1Dn. Let x \in D and W
be an open subset of Z with F (x, y0) \subseteq W. By the definition of the sets Dn, for each n, there are
neighborhoods Un of x and Vn of y0 and a finite subset \scrH n of \scrG n such that Un \times Vn \subseteq F+(
\bigcup
\scrH n)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
POINTS OF UPPER AND LOWER SEMICONTINUITY OF MULTIVALUED FUNCTIONS 1227
and F (x, y0) \cap H \not = \varnothing for each H \in \scrH n. Since Z is weakly k-developable,
\bigl\{ \bigcap
i\leq n(
\bigcup
\scrH i)
\bigr\}
is a base at F (x, y0). Hence there is some n0 \in \BbbN such that
\bigcap
i\leq n0
(
\bigcup
\scrH i) \subseteq W. Therefore\bigl( \bigcap
i\leq n0
Ui
\bigr)
\times
\bigl( \bigcap
i\leq n0
Vi
\bigr)
\subseteq F+(W ). This proves upper semicontinuity of F at each point of
D \times \{ y0\} .
In order to obtain an application for Theorem 2.1, we need to the following axillary result.
Lemma 2.1. Let F : X \times Y \rightarrow \scrP (Z) be a lower horizontally quasicontinuous function. If U
and V are open subsets of X and Y respectively such that U \subseteq A for some subset A of X. Then
F (U \times V ) \subseteq F (A\times V ).
Proof. Suppose that z is an arbitrary element of F (U \times V ) and W is a neighborhood of z. Take
some a \in U and b \in V such that z \in F (a, b). By lower horizontal quasicontinuity of F, we can find
some (a1, b1) \in U \times V and a neighborhood U1 of a such that U1 \subseteq U and F (x, b1) \cap W \not = \varnothing for
all x \in U1. Since U1 \subseteq U \subseteq A, we have U1 \cap A \not = \varnothing . Let a0 be an element of U1 \cap A, then
\varnothing \not = F (a0, b1) \cap W \subseteq F (A\times V ) \cap W.
This means that z \in F (A\times V ).
The following result gives a sufficient condition for symmetrically upper quasicontinuity of a
lower horizontally quasicontinuous function.
Theorem 2.2. Let X be a Baire space, Y a space with a countable base at y0 \in Y and Z a
normal space. Let F : X\times Y \rightarrow \scrC (Z) is a lower horizontally quasicontinuous function such that Fx :
Y \rightarrow \scrC (Z) is upper semicontinuous for each x \in X and F y0 : X \rightarrow \scrC (Z) is categorically upper
quasicontinuous. Then F is symmetrically upper quasicontinuous with respect to the second variable
at each point of X \times \{ y0\} .
Proof. Let x \in X and G be a neighborhood of x. Let F (x, y0) \subseteq W for some open subset W
of Z. By normality of Z, there is an open subset W \prime of W such that
F (x, y0) \subseteq W \prime \subseteq W \prime \subseteq W.
Since F y0 : X \rightarrow \scrC (Z) is categorically upper quasicontinuous at x, there is a subset A of G such
that A is of the second category in X and F (a, y0) \subseteq W \prime for all a \in A.
Let \{ Vn\} be a base of neighborhoods of y0. Define
An = \{ a \in A : Fa(Vn) \subseteq W \prime \} , n \in \BbbN .
It follows from upper semicontinuity of Fa : Y \rightarrow \scrC (Z) for each a \in A that A =
\bigcup \infty
n=1An.
Since A is of the second category in X, there is some m \in \BbbN such that Um = (Am)\circ \not = \varnothing . Let
U = G \cap Um, V = Vm and A0 = Am \cap U. Since Um \subseteq Am \cap Um, we have Am \cap Um \not = \varnothing . The
relation
\varnothing \not = Am \cap Um \subseteq G \cap Um = U
implies that U is a nonempty open subset of G. Since U \subseteq U \cap Am \subseteq A0, by Lemma 2.1,
F (U \times V ) \subseteq F (A0 \times V ) \subseteq W \prime \subseteq W.
This proves that F is symmetrically upper quasicontinuous at (x, y0).
The following result follows immediately from Theorems 2.1 and 2.2.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1228 A. K. MIRMOSTAFAEE
Theorem 2.3. Let X be a Baire space, Y a space with countable base at y0 \in Y and Z a
normal weakly k-developable space. Let F : X\times Y \rightarrow \scrK (Z) is a lower horizontally quasicontinuous
function such that Fx : Y \rightarrow \scrK (Z) is upper semicontinuous for each x \in X and F y0 : X \rightarrow \scrC (Z) is
categorically upper quasicontinuous. Then there is a dense G\delta subset D of X such that F is jointly
upper semicontinuous at each point of D \times \{ y0\} .
The following example shows that lower horizontal quasicontinuity of F in Theorem 2.3 is
necessary.
Example 2.1 ([19], Example 1). Let E = \{ (pn, qn) : n \in \BbbN \} be a countable dense subset of \BbbR 2
such that if (pn, qn) and (pm, qm) are two distinct elements of E, then pn \not = pm and qn \not = qm.
Define F : \BbbR 2 \rightarrow \scrK (\BbbR ) by
F (x, y) =
\Biggl\{
\{ 0\} , (x, y) /\in E,
[0, n], (x, y) = (pn, qn) \in E.
It is easy to see that all selection Fx and F y are upper semicontinuous but F has no point of joint
upper semicontinuity.
3. Lower semicontinuity of two variable set-valued functions. In this section, we discuss
about conditions which imply joint lower semicontinuity of a set-valued function F : X\times Y \rightarrow \scrC (Z)
on a set D \times \{ y0\} , where D dense subset of X and y0 \in Y.
Theorem 3.1. Let X be a Baire space, Y a space and Z be second countable. Let F :
X \times Y \rightarrow \scrP (Z) is symmetrically lower quasicontinuous with respect to the second variable at each
point of X \times \{ y0\} for some y0 \in Y. Then there is a dense G\delta subset D of X such that F is lower
semicontinuous at each point of D \times \{ y0\} .
Proof. Let \scrW = \{ Wn : n \in \BbbN \} be a countable base for Z. Define
An =
\bigl\{
x \in X : (x, y0) \in F - (Wn) \setminus
\bigl(
F - (Wn)
\bigr) o\bigr\}
, n \in \BbbN .
Let Dn = X \setminus An for each n and D =
\bigcap \infty
n=1Dn. Since each Dn is an open subset of X, D is
a G\delta subset of X. We will show that Dn is dense in X for each n \in \BbbN . Take some fixed n \in \BbbN
and a nonempty open subset U of X with An \cap U \not = \varnothing . Let x \in An \cap U. By symmetrical lower
quasicontinuity of F at (x, y0), there is a nonempty open subset U \prime of U and a neighborhood V
of y0 such that U \prime \times V \subseteq F - (Wn). Since U \prime is open An \cap U \prime = \varnothing . Hence U \prime \subseteq U \cap Dn \not = \varnothing .
Clearly F is jointly lower semicontinuous at each point of D\times \{ y0\} . Let x \in D and W be an open
subset of Z with F (x, y0)\cap W \not = \varnothing . Then there is some natural number n0 such that Wn0 \subset W and
F (x, y0) \cap Wn0 \not = \varnothing . Since x /\in An0 , we have (x, y0) \in
\bigl(
F - (Wn)
\bigr) o \subseteq
\bigl(
F - (W )
\bigr) o
. This proves
lower semicontinuity of F on D \times \{ y0\} .
It is natural to ask when a set-valued two variable function is symmetrically lower semicontinuous?
The following result gives a partial answer to this question.
Theorem 3.2. Let X be a Baire space, Y be a space with a countable base at y0 \in Y and
let Z be a regular space. Let F : X \times Y \rightarrow \scrC (Z) is an upper horizontally quasicontinuous
function such that Fx : Y \rightarrow \scrC (Z) is lower semicontinuous at y0 for each x \in X and F y0 :
X \rightarrow \scrC (Z) is categorically lower quasicontinuous. Then F is symmetrically lower quasicontinuous
with respect to the second variable at each point of X \times \{ y0\} .
Proof. Let for some x0 \in X, F is not symmetrically lower quasicontinuous with respect to
the second variable at (x0, y0). Then there are open sets U \subset X, V \subset Y and W \subset Z such that
F (x0, y0) \cap W \not = \varnothing but for every nonempty open subset U \prime \subseteq U and neighborhood V \prime \subseteq V of y0,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
POINTS OF UPPER AND LOWER SEMICONTINUITY OF MULTIVALUED FUNCTIONS 1229
there is some (x\prime , y\prime ) \in U \prime \times V \prime such that F (x\prime , y\prime ) \cap W = \varnothing . Let z \in F (x0, y0) \cap W. Using the
regularity of Z, we can choose some open subset W \prime of W such that
z \in W \prime \subseteq W \prime \subseteq W.
Since F y0 is lower categorically quasicontinuous at x0, there is a subset A of U of the second
category such that
F (a, y0) \cap W \prime \not = \varnothing , a \in A.
Let \{ Vn : n \in \BbbN \} be a countable base at y0, define
An = \{ a \in A : Vn \subseteq F -
a (W \prime )\} , n \in \BbbN .
Since for every x \in X, the function Fx is lower quasicontinuous at y0 \in Y, we have A =
\bigcup \infty
n=1An.
We will get into a contradiction by showing that each An has empty interior. Let n be a
fixed positive integer and G be an arbitrary nonempty open subset of X. By our assumption,
F (x1, y1)\cap W = \varnothing for some (x1, y1) \in G\times Vn. Therefore F (x1, y1) \subseteq Z \setminus W \prime . Since F y1 is upper
horizontally quasicontinuous, there is some nonempty open subset G\prime of G and y2 \in Vn such that
F (G\prime \times \{ y2\} ) \subseteq Z \setminus W \prime . It follows that G\prime \cap An = \varnothing .
The following result follows immediately from Theorems 3.1 and 3.2.
Theorem 3.3. Let X be a Baire space, Y a space with countable base at y0 \in Y and Z a
second countable regular space. Let F : X \times Y \rightarrow \scrC (Z) is an upper horizontally quasicontinuous
function such that Fx : Y \rightarrow \scrC (Z) is lower semicontinuous and F y : X \rightarrow \scrC (Z) is categorically
lower quasicontinuous for each (x, y) \in X \times Y. Then there is a dense G\delta subset D of X such that
F is jointly lower semicontinuous at each point of D \times \{ y0\} .
Example 3.1 ([19], Example 2). Let E be the set that was defined in Example 2.1. Define F :
\BbbR 2 \rightarrow \scrC (\BbbR ) by
F (x, y) =
\left\{ \{ n\} , (x, y) = (pn, qn) \in E,
[0,\infty ), (x, y) /\in E.
One can easily check that all selection Fx and F y are lower semicontinuous but F has no point of
joint lower semicontinuity. Therefore the assumption of upper horizontal quasicontinuity of F in
Theorem 3.3 is necessary.
Theorem 3.4. Let X be a Baire space, Y a topological space, y0 \in Y and Z a developable
space. Let F : X \times Y \rightarrow \scrK (Z) be symmetrically lower quasicontinuous with respect to the second
variable and F y0 be upper quasicontinuous. Then there is a dense G\delta subset D of X such that F
is jointly lower semicontinuous at each point of D \times \{ y0\} .
Proof. Let \{ \scrG n\} be a development for Z. For each n \in \BbbN , let An be the set of all x \in X
such that for some \scrH n \subseteq \scrG n, F (x, y0) \subseteq
\bigcup
\scrH n and (x, y0) \in
\bigl(
F - (W )
\bigr) o
for each W \in \scrH n with
(x, y0) \in F - (W ).
We will show that Dn = (An)
o is dense in X for each n \in \BbbN . Take some n \in \BbbN and let
U be an arbitrary nonempty open subset of X. Let x \in U and \scrH n be a finite subset of \scrG n with
F (x, y0) \subseteq
\bigcup
\scrH n. Since F y0 is upper quasicontinuous at x, there is a nonempty open subset U \prime of U
such that F (t, y0) \subseteq
\bigcup
\scrH n for each t \in U \prime . Let \scrH n = \{ W1, . . . ,Wk\} . If U \prime \times \{ y0\} \cap F - (W1) = \varnothing
put U1 = U \prime and V1 = Y. Otherwise, by symmetrical lower semicontinuity of F on U \prime \times \{ y0\} , we can
choose a nonempty open subset U1 of U \prime and a neighborhood V1 of y0 such that U1\times V1 \subseteq F - (W ).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1230 A. K. MIRMOSTAFAEE
By applying this method k-times, we can find a nonempty open subsets Ui of U and a neighborhood
Vi of y0 such that either Ui \times \{ y0\} \cap F - (Wi) = \varnothing or Ui \times Vi \subseteq F - (Wi), where Ui \subseteq Ui - 1 and
Vi \subseteq Vi - 1 for each 1 < i \leq k. Thus Uk \subseteq An. Since Uk is open, Uk \subseteq Dn. Therefore, each Dn is
dense in X. Let D =
\bigcap \infty
n=1Dn. Since X is a Baire space, D is dense in X. We will show that F
is lower semicontinuous at each point of D \times \{ y0\} .
Let x \in D and W be an open subset of Z with F (x, y0)\cap W \not = \varnothing . Take some z \in F (x, y0)\cap W.
It follows from the definition that for each n \in \BbbN , there is some Wn \in \scrH n \subseteq \scrG n such that
F (x, y0) \subseteq
\bigcup
\scrH n and z \in Wn. Since \{ Wi\} is a base at z, there is some n0 \in \BbbN such that
Wn0 \subseteq W. Therefore
(x, y) \in
\bigl(
F - (Wn0)
\bigr) o \subseteq \bigl(
F - (W )
\bigr) o
.
Hence F is jointly lower semicontinuous at (x, y0).
Corollary 3.1. Let X be a Baire space, Y a space with countable base at y0 \in Y and let Z be
a metric space. Let
(a) F : X \times Y \rightarrow \scrK (Z) is upper and lower horizontally quasicontinuous function,
(b) Fx : Y \rightarrow \scrK (Z) is upper and lower semicontinuous and
(c) F y : X \rightarrow \scrK (Z) is categorically upper and lower quasicontinuous for each (x, y) \in X \times Y.
Then there is a dense G\delta subset D of X such that F is jointly upper and lower semicontinuous at
each point of D \times \{ y0\} .
Proof. According to Theorems 2.1 and 3.4, there are dense G\delta subsets D1 and D2 of X such
that F | D1\times \{ y0\} is jointly lower semicontinuous and F | D2\times \{ y0\} is jointly upper semicontinuous. Since
X is Baire, D = D1 \cap D2 is a dense G\delta subset of X and F | D\times \{ y0\} is jointly upper and lower
semicontinuous.
The following result follows immediately from Theorem 3.4.
Corollary 3.2 ([21], Theorem 2). Let X be a Baire space, Y a topological space and Z a
developable space. If f : X \times Y \rightarrow Z is symmetrically quasicontinuous with respect to the second
variable, then, for each y0 \in Y, there is a dense G\delta subset D of X such that f is jointly continuous
at each point of D \times \{ y0\} .
Corollary 3.3 ([15], Theorem 2). Let X be a Baire space, Y a topological space and Z a Moore
space. If Y has a countable base in y0 \in Y and f : X \times Y \rightarrow Z is a horizontally quasicontinuous
function such that fy0 is categorically quasicontinuous and fx is continuous for each x \in X, then
there is a dense G\delta subset D of X such that f is jointly continuous at each point of D \times \{ y0\} .
Proof. By Theorem 3.2, f is symmetrically quasicontinuous with respect to the second variable
at each point of X \times \{ 0\} . So that the result follows from Corollary 3.2.
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Received 12.11.14,
after revision — 08.02.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
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| id | umjimathkievua-article-1772 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:22Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/28/a7bd77cc69511b2d2322378f7992eb28.pdf |
| spelling | umjimathkievua-article-17722019-12-05T09:26:20Z Points of upper and lower semicontinuity of multivalued functions .................. Точки верхньої та нижньої напiвнеперервностi для багатозначних функцiй Mirmostafaee, A. K. Мірмостафае, А. К. We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely. among other results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping $F : X \times Y \rightarrow \scr K (Z)$ is jointly upper semicontinuous on sets of the from $D \times \{ y_0\}$, where $D$ is a dense G\delta subset of $X$ and $y_0 \in Y$. A similar result is obtained for the joint lower semicontinuity of upper horizontally quasicontinuous mappings. These results improve some known results on the joint continuity of single-valued functions. Вивчається спiльна верхня та нижня напiвнеперервнiсть для багатозначних функцiй двох змiнних. Бiльш точно, серед iнших результатiв показано, що за деяких умов нижньо горизонтально квазiнеперервне вiдображення вiд двох змiнних $F : X \times Y \rightarrow \scr K (Z)$ є спiльно верхньо напiвнеперервним на множинах з $D \times \{ y_0\}$, де $D$ — щiльна $ G\delta $ пiдмножина $X$ та $y_0 \in Y$. Подiбний результат отримано також для спiльної нижньої напiвнеперервностi верхньо горизонтальних квазiнеперервних вiдображень. Цi результати покращують деякi вiдомi результати про спiльну неперервнiсть однозначних функцiй. Institute of Mathematics, NAS of Ukraine 2017-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1772 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 9 (2017); 1224-1231 Український математичний журнал; Том 69 № 9 (2017); 1224-1231 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1772/754 Copyright (c) 2017 Mirmostafaee A. K. |
| spellingShingle | Mirmostafaee, A. K. Мірмостафае, А. К. Points of upper and lower semicontinuity of multivalued functions .................. |
| title | Points of upper and lower semicontinuity of multivalued functions .................. |
| title_alt | Точки верхньої та нижньої напiвнеперервностi
для багатозначних функцiй |
| title_full | Points of upper and lower semicontinuity of multivalued functions .................. |
| title_fullStr | Points of upper and lower semicontinuity of multivalued functions .................. |
| title_full_unstemmed | Points of upper and lower semicontinuity of multivalued functions .................. |
| title_short | Points of upper and lower semicontinuity of multivalued functions .................. |
| title_sort | points of upper and lower semicontinuity of multivalued functions .................. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1772 |
| work_keys_str_mv | AT mirmostafaeeak pointsofupperandlowersemicontinuityofmultivaluedfunctions AT mírmostafaeak pointsofupperandlowersemicontinuityofmultivaluedfunctions AT mirmostafaeeak točkiverhnʹoítanižnʹoínapivneperervnostidlâbagatoznačnihfunkcij AT mírmostafaeak točkiverhnʹoítanižnʹoínapivneperervnostidlâbagatoznačnihfunkcij |