On feeble 7] -continuity and feeble Tj-cliquishness of multivalued maps

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COM MENT ATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATE MATY CZNEGO

Séria I. PRACE MATEMATYCZNE XXVIII (1989)

M

P

(Bialystok)

On feeble 7] -continuity and feeble Tj-cliquishness of multivalued maps

Abstract. In this paper we introduce the class of lower and upper feeble 7]-cliquish maps.

Such maps are a natural generalization of lower and upper quasi-continuous maps defined by Popa in [11]. This paper presents an investigation of the fundamental properties of lower and upper feeble 7i-cliquish maps and their relationship to other types of maps.

Let I be a topological space and let Ü be an open cover of X. The cover Ü is called T2-open (T3-open) if for every U eÜ, the interior of X \ U is not empty (there are open sets W and W such that W cz W <= W cl X \ U), where ( ) denotes the closure in X [7].

Let Ü, 0 be two collections of subsets of a space X and let К be a subset of X. We define:

|0 |и ^ Ü (resp. ^ Ü) if and only if for every pair of sets O', O" eÔ there exists a set U e Ü such that O' cz U and O" c U (resp. O' n U # 0 and O ' n t / ^ 0 ) ;

|K| < Ü if and only if for each points x', x" e К there exists a set U eÜ such that x’eU and x" e U ;

|0| ^ Ü if and only if for every set O e ô we havr |0| ^ Ü.

If F is a multivalued map from a topological space X into a topological space Y, written F: X -*Y, then for any sets A cz X, B q Y and a collection Ü of subsets of Y we write:

F +(B) = \ x e X : F(x)c:B} ([1]);

F~{B) = \ x - X : F ( x ) n B ^ 0 | ([1]);

F +(Ü) = \F+(U): U eÜ\; F~ (Ü) = \F~ (U): U eÜ) ; F (A) = \F(x): x e A ) .

D

1. A multivalued map F: X Y is said to be:

Lower feeble ^-continuous at a point x e X if for eve|ry open cover Ü of Y there exists an open set W cz X containing x such tjiat \F(W)\t < Ü ;

Lower feeble Fx -cliquish at a point x e X if for every | open cover Ü of Y and every open set W a X containing x there exists an open non-empty set G cz W such that |F(G)|,. ^ Ü ;

- Commentationes Math. 28.2

332 Mar i an Przemski

U p p e r f e e b l e ^ - c o n t i n u o u s at a point

X if for every open cover Ü of Y such that F(x)czU for some U et) there exists an open set W с X containing x such that |F(W% ^ Ü; .

Upper feeble Tx-cliquish at a point x e X if for every open cover Ü of Y such that F(x) czU for some U e(J, and for every open set W с X contain­

ing x there exists an open non-empty set G c j y such that |F (G)|„ ^ Ü.

By Cl (F) (Cj (F)) and A\ (F) (A} (F)) we denote the set of all points at which F is upper (lower) feeble Ti-continuous and upper (lower) feeble T[- cliquish, respectively.

A map F is upper (lower) feeble Tx -continuous, and upper (lower) feeble Tx- cliquish if C\ (F) = X (C] (F) = X), and A\ (F) = X (Aj (F) = X), respectively.

It is easy to see that when F(x) = !/(x)] for a single valued m ap/:

X - + Y the definitions of lower feeble 71-continuity (or upper feeble Tx- continuity) and lower feeble Tx -cliquishness (or upper feeble Ti-cliquishness) reduce to the definitions of feeble Tx-continuity and feeble Tx-cliquishness of the map / respectively [13]; a map / : X -* Y is said to be:

Feeble Tx-continuous at a point x e X if for every open cover Ü of Y there exists an open set W c= X containing x such that \f(W)\ ^ Ü ;

Feeble Tx-cliquish at a point

X if for every open cover Ü of Y and every open set W c= X containing x there exists an open non-empty set G c z W such that |/(G)| ^ Ü.

Pr o p o s i t i o n

1. I f a map F: X -> Y is upper (lower) semi-continuous at a point x e X , then F is upper (lower) feeble Tx-continuous at this point.

P roof. Assume that F is upper semi-continuous at x e X . Let Ü be an open cover of Y such that F (x) c= U for some U e(J. Then there exists an open set W <= X containing x such that F(z) c U for every z e W;

|F(W)|U ^ Ü and, consequently, F is upper feeble Tx -continuous at x. Proof of the lower feeble Tx -continuity is similar and is omitted.

It is easy to verify by arguments similar to those of the above proof that upper (lower) feeble Tx -cliquish maps are natural generalizations of upper (lower) quasi-continuous maps as defined by Popa [11]:

A map F: X -> Y is said to be upper (lower) quasi-continuous at a point x e X if for each open set V c Y such that F(x) с: V (F(x) n F ^ 0 ) and for each open set W cz X containing x there exists an open non-empty set G cr W such that F(w) с V (F (w) n V Ф 0 ) for every weG.

The following diagram illustrates the relations between these classes of maps.

F is upper feeble Tx -continuous => F is upper feeble Tx -cliquish

if. ÏÏ

F is upper semi-continuous => F is upper quasi-continuous

The relations between lower feeble Tx-continuous and lower feeble Tx-

cliquish maps are similar.

Feeble Tx -continuity and feeble Tx -cliquishness 333

The following two examples show that the-class of upper (lower) feeble 7i -cliquish maps is greater than the class of upper (lower) quasi-continuous maps.

E

1. Let X be the space of real numbers with the natural topology. Let У be the space of real numbers with the topology generated by Ü — {( — oo, 1], (1, oo)}. We shall demonstrate that the map F: X -*■ Y given by the formula

F(x) = for A = 1, for X Ф 1,

is upper feeble 7i -cliquish but is not upper quasi-continuous. The map F is clearly upper semi-continuous at each point x Ф 1. Let x = l and let À be an open cover of У. Now either У e À or À = Ü. If Y e À, then for every open set W с X containing x we have \F(W)\U ^ Â. If À = Ü, then for every open set W cz Y containing x there exists an open non-empty set G c W ( G

= W \ \ \ } ) such that F(z) cz (1, oo) for every z e G : so \F(G)\U^ A . This means that F is upper feeble 7^-cliquish at x = 1. On the other hand,

( - o o , 1 ] с У is an open set such that F (l) cz( — oo, 1] and for each open set G cz X there exists a point x 'eG \{ l} c G such that F(x') £ ( — oo, 1].

Thus, F is not upper quasi-continuous at x = 1.

E

2. Let X be the space of real numbers with the natural topology. Let У be the space of real numbers with the topology generated by Ü = {( — oo, 1], (1, oo){. By Q we denote the set of rational numbers. Define F : X У by

F(x) = for x e Q,

for x$Q.

We shall show that F is lower feeble Tt -cliquish but is not lower quasi- continuous. At first let x e Q and let À be an open cover of У Now either У EÀ or À = Ü. If У EÀ, then for every open set W с. X containing x we have |F(W% ^ À. If À = Ü, then for every open set G а X and for any

x ' , x " e G we have F(x') n ( — oo, 1] Ф 0 and F(x") n ( — oo, 1] Ф 0 ; so

|F(G)|, < A. Secondly, if x<£Q, then by arguments similar to those above F is lower feeble 7i-cliquish at x. Thus, F is lower feeble 7i-cliquish at every point. Clearly, F is not lower quasi-continuous at every point x e Q. Indeed;

V = (1, oo) с У is an open set such that F(x) n V Ф 0 and for each open set G cz X there exists a point x ' e G\Q cz G such that F(x') n V = 0. Thus, F is not lower quasi-continuous at x.

D

2 [4]. A map F from a topological space X into a uniform space У with a uniformity is said to be cliquish at a point x e X if for every V e jU and for every open set W cz X containing x there exists an open non-empty set G cz W such that [F(x') xF(x")] n K / 0 for any x', x" e G.

A map F is cliquish if it is cliquish at every x e X.

334 Mar i an Prz emski

Let ^ be a uniformity induced by a metric on Y. I f / : X Y is a single valued map and F (x )= [/(x )} , then the cliquishness of F means the cliquishness of / [2], [6], [8], [10], [14].

Simultaneously, we consider a uniform space Y with a uniformity 41 as a topological space with a topology induced by the uniformity 41. For any

x e

Y and V

4J we write B(x, V) = \y

Y : (x, y)

V } and C{V)

= \B(x, V ):

Y } .

P

2. Let Y be a uniform space with a uniformity 41. I f a map F: X -> Y is lower feeble 7]-cliquish at a point x e X, then F is cliquish at this point.

We use the following result in the proof of Proposition 2.

L

1. Let Y be a uniform space with a uniformity W. A map F: X

—► Y is cliquish at a point x e X if and only if for every V

4/ and for every open set W <= X containing x there exists an open non-empty set G cr W such that |F(G )|,<C (F).

Proof. Assume that F is cliquish at a point x. Let V e 4/ and let W a X be an open set containing x. There exists an open non-empty set G c: W such that [ F ( x ') x F ( x '') ] n F ^ 0 for any x',x"eG. Thus for any F(x'), F(x") e F{G) there exist points a ' e F (x1), a" e F( x ") such that (a',a")EV. It implies a'EB(a", V); so F(x') n B(a”, V) Ф 0 and F(x" )nB (a ", V ) Ф 0 . This means that \F(G)\, ^ C(V).

Conversely, let V e 41 and let W с X be an open set containing x. There exist U

41 and open non-empty set G <=. W such that U2 a V and

|F(G)|, ^ C { U ) . Then for each points x', x"eG we have F(x') n B (y , U ) Ф 0 ,

F ( x " ) г л В ( у , U ) Ф 0 for some B(y, U )

C ( U ) . Thus for any x', x "

G there exist points a'eF(x'), a”eF(x") such that a', a " eB(y, U ) ; so ( a ' , y ) E U ,

(ia " , y ) E U and, consequently, (a', a")

U2

V. This means that

(a', a") e [F (x') x F (x")] n V and F is cliquish at x.

P ro o f of P r o p o s itio n 2. Let V

41 and let W cz X be an open set containing x. Since the collection Ü = (Int(B(y, V)): y eT } is an open cover of the topological space Y with the topology induced by 41, by the lower feeble Tt -cliquishness of F there exists an open non-empty set G c W such that |F(G)|, ^ Ü. Clearly, it implies |F(G)|, ^ C(V). Thus by Lemma 1, F is cliquish at x and the proof is complete.

From Proposition 2 follows immediately:

C

1 ([4], Proposition 4). If a map F : X -* Y is lower quasi- continuous at a point x, then F is cliquish at this point.

The converse of Proposition 2 is not true as the next example shows.

E

3. Let R be the space of real numbers with the natural metric

Feeble T{-continuity and feeble ^-cliquishness

and let Y — (0, oo) <= R have the usual subspace topology. Let us consider the set X = [0, oo) c R with the topology T = J0, X} и {(r, oo), X: r > 0}.

Let an = (3n + 4)/(3(n+ l)(n + 2)), bn = (Зи + 2)/(Зи(и + 1)) for n = 1, 2, ...

By Q we denote the set of rational numbers. Define F: X -+ Y by

F(x) =

(bi, oo) (b2, !) (b„, a„_2) '{bn+ 2» &n)

for x e[0, 1) n Q, for x e [ l, 2) n Q,

for x e [n — 1, n) n Q , n = 3 ,4 , . . . , for x e [ n — 1, n)\Q, n = 1 ,2 ,...

The map F is clearly cliquish. We shall demonstrate that F is not lower feeble 7j -cliquish. Note that the collection Ü = \(an, bn):

и = 1, 2, ...} u !(|, oo)[ is an open cover of Y and for every open non­

empty set G ci X there exist points x', x"eG such that for every set U e(J either Fix') n U Ф 0 or F(x") n U = 0 . Indeed, if G ci X is an open set, then there exists a number n > 3 such that [n — 1, n) c G and x 'e [ n — 1, ri)nQ, x" e [ n — 1, n)\Q; we have F(x') = (b„, an- 2) c («п- ь ^n-i) an^ F(x")

= (b„+2, u j c ( a B+1, bn+l)eÜ. Moreover, («„_!, b„_i) п (аи+1, Ья+1) = 0 and for every set U eÜ such that U Ф (a,,-!, b„_!) and U Ф (an+l, b„+1) we have F (x ')n G = 0 and F ( x ') n U = 0 . Thus, F is not lower feeble Tx -cliquish.

P

3. Any cliquish map F from a topological space X into a uniform compact space Y is lower feeble f-cliquish.

P roof. Let x e X and let W a X be an open set containing x. Let Ü be an open cover of Y. Since Y is compact, there exists a set K e t such that for every set A sC(V) there exists a set U GÜ such that A czU. Thus, since F is cliquish, by Lemma 1 there exists an open non-empty set G <= W such that

|F(G)|| ^ C(F) and, consequently, \F(G)\i^Ü. So, F is lower feeble Tx- cliquish.

The following two results give simple characterization of upper (lower) feeble 7i-continuous and upper (lower) feeble -cliquish maps.

P

4. (i) A map F: X -> Y is lower feeble Tj-continuous (lower feeble Tx-cliquish) if and only if for every open cover Ü of Y there exists a collection À of open subsets of X such that \À\ ^ F~ (Ü) and X = {JÀ (X = \JÂ).

(ii) I f a space Y has an Tr open cover, i = 2, 3, then a map F: X -»• Y is lower feeble f-continuous (lower feeble Tj-cliquish) if and only if for every Tr open cover Ü of Y there exists a collection A of open subsets of X such that

\A\ ^ F~ (Ü) and X = U A (X = (J7).

P ro o f of (i) for lower feeble ^-continuity. Assume that F is lower feeble

-continuous and let Ü be an open cover of Y. Then for every x e X there

336 Ma r i a n Prz emski

exists an open set W(x) с X containing x such that |F(JT(x))|/ < Ü. Thus for each points x ', x " e W ( x ) we have F ( x ' ) n U ^ 0 and F ( x " ) n [ / # 0 for some UeÜ. So we have x', x" eF~ (U) eF~ (Ü) and this means that

\W(x)\ ^ F~ (Ü). Let us put À = \ W(x): x e l j . We see that \À\ ^ F~ (Ü) and (J À’ = X.

Conversely, let x e X and let Ü be an open cover of Y If A is a collection of open subsets of X such that \Â\ ^ F~ (Ü) and À covers X, then there exists a set A e  containing x such that \À\ ^ F~ {Ü). Thus for each points x', x” eA we have x', x” eF~ (U) for some U eÜ. So F ( x ') n U Ф 0 and F(x”) n U Ф 0 , i.e., |Т(Т)|г ^ Ü. This means that F is lower feeble Tx- continuous.

For lower feeble Tx -cliquishness the proof is analogous and is omitted.

P ro o f of (ii) for lower feeble 7Ï-cliquishness. Necessity follows immedi­

ately from (i).

To prove the converse, we suppose that for every T2-open cover Ü of Y there exists a collection A of open subsets of X such that \À\ ^ F~ (Ü) and X = y À. Let x e X and let 0 be an open cover of Y. By the assumption there exists an T2-open cover O' of Y It is easy to verify that the collection Ü —{On O': OeO, O'eO'} is a T2-open cover of Y. If À is a collection of open subsets of X such that \À\ ^ F~ (Ü) and X = {J A, then for every open set W c= X containing x there exists an open non-empty set A e  such that G = W n А Ф 0 , since x g y A. It is obvious that |G| ^ F~ (Ü). Then for each points x \ x"eG there exists U = O nO ' eÜ such that F(x')r\,U Ф 0 and F ( x " ) n t / # 0 , where OeO and O'eO'; so Р ( х ' ) п О Ф 0 and F(x") n O Ф 0 for some OeO. This means that |F(G)|/ ^ 0 and F is lower feeble Ti-cliquish at x.

The analogous proof for i = 3 we omit.

The simple proof for lower feeble 7i-continuity we also omit.

P

5. (i)

map F: X -> У is upper feeble T^-continuous ( upper feeble Tx-cliquish) if and only if for every open cover Ü of Y there exists a collection Л of open subsets of X such that \Â\ < F + (Ü) and [j F + (Ü) = y À (\JF+(Ü) = \JÀ).

(ii) Let F: X -» Y be d map such that for every x e X there exists a Tr open cover Ü of Y such that F(x) a U for some U eÜ, i = 2, 3. Then F is upper feeble Tt -continuous (upper feeble T1-cliquish) if and only if for every Tr open cover Ü of Y there exists a collection À of open subsets of X such that

\A\ ^ F + (Ü) and l ) F + (Ü) = U À (U F +(Ü) - ЦД).

P r o o f of (i) for upper feeble 7]-cliquishness. Assume that F is upper Ty cliquish and let Ü be an open cover of Y. If \J F + (Ü) = 0 , then clearly the proof is complete.

Assume that \JF" (Ü) Ф 0 \ then there for every xe{J F* (Ü) and every

Feeble Ti -continuity and feeble 7i -cliquishness

open set W <= X containing x there exists an open non-empty set G a W such that |F(G)|U ^ Ü. Thus, for each points x \ x" e G, we have F(x') a I f and F(x") cz V for some U’eÜ; so x', x" e F + (IF), i.e., |G| < F+ (Ü). Let us put À(x, W) = {G c W: G is open, |F(G)|„ ^ Ü], À(x) = (J U'(x, W)'- W is an open set containing x}, and À — U (Jf(x): x e (J F + (Ü)}. Clearly,

|J*| ^ F + (£/) and \JF+ (Ü) cz (J A. We see also that for every x e ( J F + (Ü) we have {x} и (J T(x) c: (JF+ (Ü), then (J Â cz (J F + (Ü) and, consequently,

u F +(Ü)= ÎJÀ.

Conversely, let x e X and let Ü be an open cover of Y such that F(x) c U for some U eÜ. Assume that A is a collection of open subsets of X such that \Â\ < F + (Ü) and U A — (J F + (Ü). Then for every open set W cz X containing x there exists an open set A' e À such that W z z G = W r ^ A ' ^ 0 . Since \A'\ ^ F + (Ü), if follows that for each points x', x"eG c A' we have x', x " e F + (IF) for some IF e I7; so |F(G)|U ^ Ü. Thus, F is upper feeble Tt- cliquish.

The proof for upper feeble Tx -continuity is omitted, because it is quite similar to the above proof.

P ro o f of (ii) for upper feeble Tx-continuity. Necessity follows immedi­

ately from (i).

To prove the converse we assume that for every T2-open cover Ü of Y there exists a collection À of open subsets of X such that \Â\ ^ F + (Ü) and (J F + (C) = (J T. Let x e X and let 0 be an open cover of Y such that F(x) czO for some O e O. By the assumption, there exists a T2-open cover O' of Y such that F(x)czO' for some O’ e O’. Clearly, the collection Ü

= { 0 n O': О

O, O'

O' ) is a T2-open cover of У such that

(J F + (Ü). Let Л be a collection of open subsets of X such that \A\ ^ F + (Ü) and (J F + (Ü)

= (JX Then there exists an open set A e A containing x. Clearly,

|F(^4)|M ^ Ü, since \A\ ^ F + (Ü). Thus, F is upper feeble Ti-continuous at x.

The analogous proof for i = 3 we omit.

For upper feeble Tx -cliquishness the proof is analogous and‘is omitted.

If F: X -> У is a multivalued map, then for every open cover Ü of У we denote:

N(Ü) = Ix e Z : for every U EÜ we have F(x) <p. U)\

GU(Ü) = |x e X : |F(IF)|U ^ Ü for some open set W containing x] ; Gi(Ü) = JxeZ : \F(W)h ^ Ü for some open set W containing x ].

It is easy to verify that Gu(Ü) and Gt(Ü) are open sets. We see also that GU(Ü) cz Gi(Ü) and Gu(Ü)nN(Ü ) = 0 for every open cover Ü of У

P

6. For a multivalued map F: X -* Y we have :

(i) Cl(F) = П {N(Ü) u GU(Ü): Ü is an open cover of У},

338 Mar i an Prz emski

(ii) Al{F) = 0 {N(Ü) GU{Ü): Ü is an open cover of Y}, (iii) Cj(F) = (~){Gi(Ü): Ü is an open cover of Y),

(iv) Af(F) = П {£/(£/): Ü is an open cover of Y}.

P roof. The simple proofs of (i) and (iii). are omitted.

We will show (ii). Assume that F is upper feeble Tx -cliquish at a point x, i.e., xeAl(F). Let Ü be an open cover of Y Then either x eN (Ü ) or x$N(Ü). Assume that хфЫ(0), i.e., F(x) c= U for some U e Ü . By the upper feeble Tx -cliquishness of F at the point x, for every open set W cz X containing x there exists an open non-empty set K cz W such that

|F(/C)|„ ^ Ü. We observe that K cz GU(Ü); so x e G u(Ü). This means that for every open cover Ü of Y we have x e N (Ü )u GJV).

Conversely, assume that for every open cover1 Ü of Y we have x eN(Ü) u Gu(0); we shall prove that F is upper feeble Tx-cliquish at x. Let Ü be an open cover of Y such that F(x) c: U for some U eÜ. So x e G u{0), and for every open set W cz X containing x there exists an open non-empty set K cz W such that K cz GU{Ü), since the set GU(Ü) is open. Consequently, by the definition of GU(Ü), there exists an open non-empty set G <= K cz W

such that |F(G)|„ ^ Ü. Hence x eA l(F ) and the proof of (ii) is complete.

Although the proof of (iv) is not identical to the above proof it is quite similar and is omitted.

C

2. For each map F: X -+ Y the set Al(F) is closed.

C

3. I f for a map F: X -► Y the set C\ (F) is dense in X, then F is lower feeble Tx -cliquish.

P ro o f. It follows by:

X = Cj(F) = П \Gi(Ü): Ü is an open cover of Y)

cz fj \Gi(Ü): Ü is an open cover of Y} = A}(F).

C

4. I f F is a multivalued map of a Baire space X into a topological space Y and the set X \ C j ( F ) is of the first category, then F is lower feeble Tx-cliquish.

We denote by CU(F) and Ct(F) the set of upper and lower semi­

continuity points of F, respectively.

The following result improves Theorem 7 of [4].

C

5. I f F is a multivalued map of a Baire space X into a uniform space Y and the set X \ C t(F) is of the first category, then F is lower feeble Tx- cliquish.

P roof. If the set X \ C t(F) is of the first category, then the set X\Cj{F)

Feeble T^-continuity and feeble Тг-cliquishness 339 is also of the first category, since C,(F) cz C j ( F) . Thus, by Corollary'4, F is lower feeble 7}-cliquish.

We say that a topological space X is a nearly Tx-space at a point x e X , if for every open set W cz X containing x there exists an open cover Ü of X such that St(x, Ü) czW [12], where St(x, Ü) denotes the star of the point x with respect to Ü.

It is easy to verify that nearly 7] is strictly weaker than 7].

L

2. I f F: X -> Y is a map such that for every point x e X there exists a point zeF(x) such that Y is a nearly Tx-space at z, then we have C„ (F) = CU(F).

P roof. Let xeCl(F) and let V c Y be an open set such that F(x) cz V If Y is nearly Tx at z eF(x), then there exists an open cover Ü of Y such that St(z, Ü) cz V By the upper feeble 7]-continuity of F at x, there exists an open set W cz X containing x such that \F(W)\U ^ 0 = {K}uÜ. Thus for every p e W there exists a set O e ô such that F(x) c= О and F(p) cz O. Since V is the element of 0 such that St(z, Ü) cz V, we see that if F(x) c z O e Ü ,

then O cz St(z, Ü) cz V So F(p) с: V for every p e W and this means that F is upper semi-continuous at x, i.e., x e C u(F). Thus, by Proposition 1, we have Clu(F) = Cu(F).

For a multivalued map F : X -> Y we denote by C*(F) the set of all points x e X at which the following condition is satisfied:

For every open set V cz Y such that F(x) cz V there exists an open set W cz X containing x such that F(x') r\V Ф 0 for any x' e W [4], p. 84.

L

3. / / F: X -*■ Y is a map with closed values, then we have

■Cl(F)czC^F).

P ro o f. Let x e C l ( F ) and let V cz Y be an open set such that F(x) cz V Let Ü — \V, Y\F(x)}. Tfien there exists an open set W cz X containing x such that \F (W )h^ Ü . Since V is the only element of Ü such that F{x)r\V Ф 0 , F(x')c\V Ф 0 for any x 'e W ; so xeC^(F).

Since for a single valued map f : X -* Y we have Cu(f) — C ,(/)

= C*(f) = C(f), where C (f) is the set of the continuity points of/, Lemmas 2 and 3 imply the following corollary.

C

6. Any feeble Tx-continuous single valued map into a nearly Tx-space is continuous.

We say that a topological space X has the property P, if there exists a sequence \Ün: n = 1 ,2 ,...} of open covers of X such that:

If À is an open cover of X, then there exists a number n' such that for

every U e Ü n., U cz A for some A e À .

340 Mar i an Prz emski

A topological space X has the property Pu if there exists a sequence

\(Jn: n = 1, of open covers of X such that:

If À is an open cover of X and A g A, then there exists a number ri such that A czU' for some U' e and for every U e Ü n., U cz В for some B g Â.

It is easy to see that the property Pt is weaker than the property Pu. The following example shows some topological space with the property Pu, and shows that the class of spaces with the property Pt is greater than the class of spaces with the property Pu.

E xample 4. (i) Let X be the space of real numbers with the topology T = {( — oo, x): x e l } . For any и е ]1 ,2 , ...] let us put £/„={( — oo, k + (l/nj): к is an integer]. If À is an open cover of X and A = ( — oo, a)eÀ for some a e X , then there exists an integer к such that ( — oo, a) c=(— oo, k + {\/ri)) for some ri. We see also that for every U g Ü„>, there exists* B g À such that U с: B. Thus, {X, T) has the property Pu.

(ii) Let X be the space of real numbers with the topology generated by Ü = \(k, k + 1]: к is an integer}. The space X has clearly the property Pt. We shall show that X is not a space with the property Pu. We shall demonstrate that for every open cover À of X there exists an open cover 0 of X and

O

O such that either O f A for every A G  , or there exists A' G  such that A' <f О for every 0 g O. At first, let À — Ü. Then 0 = Ü u !( —1, 1]], 0 = (—1, 1 ~\ g 0, and 0 £ A for every A g Â. Secondly, let À ^ Ü ; then either X G or ХфЛ. If X g À then for Ô = Ü we have A' = X g À and A' f О for any 0 g O. If X фÀ then for Ô = À kj jJA] we have X = О g O and О £ A for any A g A. Thus, X is not a space with the property Pu.

The following lemma follows trivially from the definitions of the proper­

ty Pb Pu and from Proposition 6.

L emma 4. Let F : X -* Y be a multivalued map. Then:

(i) if the space Y has the property Ph then there exists a sequence {t)n: n

= 1, 2, ...} of open covers of Y such that Cf(F) = (~){Gi(Ün): n = 1, 2, ...}

and A\ (F) = n \ G d ü J - n = l , 2 , ...};

(ii) if the space Y has the property Pu, then there exists a sequence {Ün: n

= 1, 2, ...} of open covers of Y such that

CÏ(F) = 0 { N ( Ü n) u G u(ÜJ: « = 1 ,2 ,...}

and

Al (F) = t ) \ N ( Ü n) u G u(Ün): n = 1, 2, ...}.

From the above lemma we immediately obtain:

C orollary 7. Let F: X -> Y be a multivalued map. Then : (i) if the space Y has the property Ph then the set C\ (F) is Gô;

(ii) if the space Y has the property Pu, then there exists a Gô set В с X such that B <= C\ (F).

$

Feeble Tj -continuity and feeble Tx -cliquishness 341

P roposition 7. Let X be a Baire space and let Y be a topological space with the property Pt. A map F : X -* Y is lower feeble ^-cliquish if and only if the set X \ C j ( F ) is of the first category.

Proof. If F is lower feeble 7}-cliquish, then by Lemma 4, X\Cj(F) = П №(£/„): n = 1, 2, n = 1, 2, ...}

^ U M \ G , ( ^ ) : и = 1,2, . . . }

and since for every number n the set G,(C/„) is open, the set Gi(Ün)\Gt(Ü„) is ne where dense and hence the set X\Cl{F) is of the first category.

The inverse implication follows directly from Corollary 4.

The above proposition and Corollary 6 imply the following result for single valued maps:

C orollary 8. A single valued map f of a Baire space X into a nearly Tx- space Y with the property Pt is feeble 7} -cliquish if and only if the set X \ C { f ) is of the frst category.

Since a single valued map / of a Baire space X into a mertic space is cliquish if and only if the set X \ C ( f ) is of the first category [6], [10], we have from Corollary 4 the following result:

C orollary 9. Any cliquish single valued map of a Baire space into a metric space is feeble Tx -cliquish.

Proposition 7 and Lemma 3 imply:

C orollary 10. Let X be a Baire space and let Y be a topological space with the property P,. I f a map F : X -> Y with closed values is lower feeble 7}- cliquish, then the set X\C^.(F) is of the first category.

From Lemma 4 we obtain:

C orollary 11. I f Y is a topological space with the property Ph then for every map F: X -> Y with closed values the set A} (F)\C* (F) is of the first category.

P roposition 8. I f a space Y has the property Pu, then for every map F : X ->Y the set Al(F)\Cl(F) is of the first category.

P roof. By Lemma 4, we have Al(F)\Cl{F)

= n { N ( Ü J u G „ ( Ü J : n = l , 2 ) . . . } \ n { W ( Ü J u G . ( Ü J : « = 1,2,... }

^ U {Gu(Cf»)\G„(C?^: « = 1,2, ...}.

342 Mar i an Prz emski

Applying arguments analogous to those in the proof of Proposition 7 we can show that the set Al(F)\Cl(F) is of the first category.

P roposition 9. Let F be a map from a Baire space X into a topological space Y with the property Pu and let F satisfy the following condition:

(r) for every open cover Ü of Y we have X = {J F +(Ü).

Then F is upper feeble Tx-cliquish if and only if the set X \C* (F) is of the first category.

P ro o f. Assume that the set X \ C l ( F ) is of the first category. Thus the set Cl(F) is dense since X is a Baire space, and by Lemma 4, we have

X = ClH(F) = r){N(Ün) v G u(Ün): n = 1, 2, . . . } с П { Щ ) и С Д : и = 1 , 2 , . . . } .

We see that by (r), for every number n the set N(Ün) is empty. So X czO\Gu(Ün): и = 1,2, ...}czAl{F).

The inverse implication follows directly from Proposition 8.

A multivalued map F: X -> Y is said to be barely upper (lower) semi- continuous if for every non-empty closed set M cz X the restriction F/M has at least one point of the upper (lower) semi-continuity.

When F(x) = \f(x)) for a single valued map / : X -* Y, the above definitions reduce to the definition of barely continuity of / [9].

P roposition 10. I f F: X -> Y is a barely upper semi-continuous map satisfying (r), then F is upper feeble Tx-cliquish.

P roof. Let x e X and let Ü be an open cover of У such that F(x) cz U for some U e Ü. If W cz X is an open set containing x then, by the barely upper semi-continuity of F, there exists a point z e W of the upper semi­

continuity of F/W and, clearly, F/W is upper feeble ^-continuous at z. By (r), F ( z ) a U ' for some U'et). Then there exists an open set W с X containing z such that \F(Wr n W)\u < ^{Ü. Thus} G = W n W cz W is an open non-empty set such that |T(G)|U ^ Ü. This means that F is upper feeble Tx- cliquish at x.

Since for every single valued map / : X ->Yt *F(x) = !/(x)j is a multi­

valued map satisfying (r), we have:

C orollary 12. Any barely continuous single valued map f : X ^ Y is feeble Tx -cliquish.

C orollary 13 ([3], Theorem 1.1). Any barely continuous map into a

uniform space is cliquish.

Feeble Tx -continuity and feeble 7j -cliquishness 343

P

11. Any barely lower semi-continuous map F: X -> Y is lower feeble Tt -cliquish.

P ro o f. Let x e X and let Ü be an open cover of Y. If W с X is an open set containing x, then by the barely lower semi-continuity of F there exists a point z e W of the lower semi-continuity of F/W and, consequently, F/W is lower feeble ^-continuous at г. Then there exists an open set W a X containing z such that \F(W' n W)h ^ Ü. Thus G = W n W c: W is an open non-empty set such that |F(G)|; ^ Ü. This means that F is lower feeble Tr cliquish at x.

From the above proposition and from Lemmas 3 and 4 we obtain:

C

14. I f F : X -> Y is a barely lower semi-continuous map with closed values into a space Y with the property Ph then the set X\C^.{F) is of the frst category.

A topological space X is said to be a Baire space in the narrow sense if every closed subspace of X is a Baire space [5].

P

12. I f a map F: X -> Y is barely lower semi-continuous, then for every non-empty closed set M <= X the map F/M is lower feeble 7i-cliquish.

Proof. If F is barely lower semi-continuous and M c l is a closed non-empty set, then F/M is also barely lower semi-continuous; so by Proposition 11, the map F/M is lower feeble Tx-cliquish.

P

13. Let F be a multivalued map with closed values from a Baire space in the narrow sense X into a topological spac# Y with the property Pj. I f for every non-empty closed set M с X the map F/M is lower feeble cliquish, then for every non-empty closed set M cz X the set C* {F/M) is non­

empty.

Proof. It follows by Corollary 10.

Combining Propositions 12 and 13 and Corollary 8 we obtain the following characterization of bare continuity of single valued maps.

C

15. A single valued map f from a Baire space in the narrow sense X into a nearly T^-space Y with the property Pt is barely continuous if and only if for every closed non-empty set M cz X the restriction f / M is feeble Tx-cliquish.

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