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On Continuity Properties of the Modulus of Local Contractibility
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
216, 86]93 Ž1997.
AY975660
On Continuity Properties of the Modulus of Local Contractibility Dusan ˇ Repovs* ˇ Institute of Mathematics, Physics, and Mechanics, Uni¨ ersity of Ljubljana, P.O. Box 2964, 1001, Ljubljana, Slo¨ enia
and Pavel V. Semenov † Moscow State Pedagogical Uni¨ ersity, Ul. M. Pyrogo¨ skaya 1, Moscow, 119882, Russia Submitted by Ying-ming Liu Received February 6, 1997
Let M X be the set of all metrics compatible with a given topology on a locally contractible space X and let for each triple z s Ž r , x, « . g M X = X = Ž0, `., DŽ z . be the set of all positive d such that the open d-neighborhood of x is contractible in the open «-neighborhood of x in metric r . We prove several continuity properties of the map D : M X = X = Ž0, `. ª Ž0, `. and then, using a selection theorem for non-lower semicontinuous mappings, show that D admits a continuous singlevalued selection. Similar, but somewhat different properties are also demonstrated for the modulus D n of local n-connectedness. Q 1997 Academic Press
1. INTRODUCTION A topological space X is said to be locally contractible if for every point x g X and for each of its neighborhoods U ; X there exists a neighborhood V ; X of x such that the inclusion V ; U is homotopically trivial. For a metric space X the notion of local contractibility admits a definition via the real-valued parameters, namely the radii of neighborhoods U and * Supported in part by the Ministry of Science and Technology of the Republic of Slovenia Grant J1-07039r94. E-mail address: [email protected]. † Supported in part by the International Science G. Soros Foundation Grant NFU000. 86 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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V. More precisely, let M X be the set of all metrics on X compatible with the given topology on X. The space M X is equipped with topology induced by the following metric of uniform convergence: dist Ž r , d . s sup min < r Ž x, y . y d Ž x, y . < , 1 4 < x, y g X 4 . For every triple Ž r , x, « . g M X = X = Ž0, `. we define DŽ r , x, « . as the set of all positive numbers d such that the d-neighborhood B Ž r , x, d . of the point x in metric r is contractible in the «-neighborhood B Ž r , x, « . of the point x in the same metric r . In this way, we define a multivalued mapping D : M X = X = Ž0, `. ª Ž0, `. with nonempty convex values which we call the modulus of local contractibility of the space X. The definition of modulus D n of local n-connectedness is very similar to the definition of modulus of local contractibility }one only needs to replace the contractibility of the ball B Ž r , x, d . inside the ball B Ž r , x, « . with the contractibility inside the ball B Ž r , x, « . of every continuous image of the n-sphere S n lying in B Ž r , x, d .. The goal of the present paper is to investigate some continuity type properties of D. As a corollary we prove the existence of a continuous singlevalued choice Žselection. of elements of the sets DŽ r , x, « . Žsee Theorem 1.5.. EXAMPLE Ž1.1.. Let X be the half-open interval w0, 1. and let r be the standard metric on R. Then DŽ r , 0, 1. s Ž0, `. and DŽ r , 0, « . s Ž0, « x, for all « - 1. This example shows that the map D does not need to be lower semicontinuous and hence in general, the standard Michael selection theory techniques do not apply. However, for a locally compact space X it is always possible to find a lower semicontinuous selection of the map D: THEOREM Ž1.2.. Let X be a locally contractible and locally compact metrizable space. Define the map = : M X = X = Ž0, `. ª Ž0, `. as follows: let for each triple Ž r , x, « . g M X = X = Ž0, `. = Ž r , x, « . s d g D Ž r , x, « . ¬ the closure of B Ž r , x, d . is compact 4 . Then the map = is lower semicontinuous. In general we can only prove the quasi lower semicontinuity of the closure of the modulus of local contractibility D. We denote by d 0 Ž r , x, « . s sup DŽ r , x, « . and DŽ r , x, « . s Ž0, d 0 Ž r , x, « .x. Clearly, DŽ r , x, « . is the closure of the set DŽ r , x, « . in the complete metric space R* s ŽŽ0, `. j `4 , c ., where c Ž t , s . s < ty1 y sy1 <
and
c Ž t , ` . s ty1 .
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THEOREM Ž1.3.. Let X be a locally contractible Ž locally n-connected. metrizable space. Then the map D : M X = X = Ž0, `. ª R* Ž respecti¨ ely, the map D n . is a quasi lower semicontinuous mapping into the complete metric space with closed, con¨ ex ¨ alues. THEOREM Ž1.4. ŽGutev w5x.. Any quasi lower semicontinuous closed ¨ alued mapping from a topological space into a complete metric space admits a lower semicontinuous closed-¨ alue selection. THEOREM Ž1.5.. Let X be a locally contractible Ž locally n-connected. metrizable space. Then there exists a single¨ alued continuous function dˆ : M X = X = Ž0, `. ª Ž0, `. such that for e¨ ery point Ž r , x, « . g M X = X = Ž0, `., the neighborhood B Ž r , x, dˆŽ r , x, « .. is contractible in the neighborhood B Ž r , x, « . Ž respecti¨ ely, e¨ ery continuous mapping of the n-sphere S n into B Ž r , x, dˆŽ r , x, « .. is null homotopic in B Ž r , x, « ... Having the values of the multivalued mapping D as convex subsets of R one can try to use the well-known Deutsch]Kenderov approach w3x. More precisely, it is natural to start by proving almost lower semicontinuity or 2-lower semicontinuity of D or D Žsee w3x for definitions.. Unfortunately, their selection theorems work only for compact convex-valued mappings. A theorem from w6x generalizes the Deutsch]Kenderov theorem, but only for closed convex-valued mappings into R. So, we chose instead the Gutev selection theorem w5, Theorem 1.4x.
2. PRELIMINARIES Recall that a multivalued map G : X ª Y is said to be a selection of a multivalued map F : X ª Y if GŽ x . ; F Ž x ., for every x g X. A multivalued map F : X ª Y is said to be lower semicontinuous if for every open subset V ; Y, the set Fy1 Ž V . s x g X ¬ F Ž x . l V / B4 is open in the space X. For a metric space Ž Y, r ., the lower semicontinuity of F at a point x can be reformulated as follows: for each positive g the following implication holds: y g F Ž x . « x g Int Ž Fy1 Ž B Ž r , y, g . . . . DEFINITION Ž2.1. w4, 5, 10x. A multivalued mapping F : X ª Y of a topological space X into a metric space Ž Y, r . is said to be quasi lower semicontinuous at a point x g X if for each positive g and for each neighborhood V Ž x . there exists a point q Ž x . g V Ž x . such that the follow-
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ing implication holds: y g F Ž q Ž x . . « x g Int Ž Fy1 Ž B Ž r , y, g . . . . Quasi lower semicontinuity of F : X ª Y means that F is quasi lower semicontinuous at every point x g X. Clearly, quasi lower semicontinuity follows from the lower semicontinuity: it suffices to put q Ž x . s x. However, the converse does not hold. Let us consider an example: EXAMPLE Ž2.2.. Let a function f : R ª R* be monotone and increasing. Then the map F : t ¬ Ž0, f Ž t .x is quasi lower semicontinuous. In this example it is easy to construct a lower semicontinuous selection of F in a straightforward manner: it suffices to put H Ž t . s Ž0, f Ž t y 0.x ; F Ž t .. We can apply the standard Michael selection theorem w8, Theorem 3.1Z x to the map H and find a singlevalued continuous selection for H and hence for F. In particular, such a selection exists for the modulus of the uniform local contractibility of a metric space Ž Y, r ..
3. PROOFS OF THEOREMS Proof of Theorem Ž1.2.. Let us check the lower semicontinuity of the map = at a point Ž r , x, « . g M X = X = Ž0, `.. To this end, we fix d g =Ž r , x, « . and some of its s-neighborhoods Ž d y s , d q s .. We need to find a neighborhood V of the point Ž r , x, « ., such that for all Ž r X , xX , « X . g V, the sets =Ž r X , xX , « X . intersect with the interval Ž d y s , d q s .. If d y s F 0, then the sets =Ž r X , xX , « X . intersect with the interval Ž d y s , d q s . for all triples Ž r X , xX , « X .. In the case when d y s ) 0 we choose numbers a and b so that
dys-b-a-d
Ž 1.
and we fix a homotopy H : B Ž r , x, d . = w 0, 1 x ª B Ž r , x, « . which contracts the neighborhood B Ž r , x, d . into a point. Let us consider the image H ŽB Ž r , x, a . = w0, 1x., where B denotes the closure of B. Because of the compactness of the set B Ž r , x, d . this image lies in the open ball B Ž r , x, « . with some ‘‘freedom.’’ More precisely, there exists a number 0 - l - « such that H Ž B Ž r , x, a . = w 0, 1 x . ; B Ž r , x, l . .
Ž 2.
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Fix a number m such that l - m - « and put r s min Ž a y b . r2, Ž m y l . r2, 1 4 .
Ž 3.
Let V1 be the r-neighborhood of the metric r in Ž M X , dist., and let V2 be the r-neighborhood of the point x in metric r and let V3 s Ž m , `. be the neighborhood of the number « . Then by the triangle inequality B Ž r X , xX , b . ; B Ž r , x, a .
Ž 4.
B Ž r , x, l . ; B Ž r X , xX , m .
Ž 5.
and
for all Ž r X , xX . g V1 = V2 . Hence for each triple Ž r X , xX , « X . g V s V1 = V2 = V3 , we have that Ž4 .
H Ž B Ž r X , xX , b . = w 0, 1 x . ; H Ž B Ž r , x, a . = w 0, 1 x . Ž2 .
Ž5 .
; B Ž r , x, l . ; B Ž r X , xX , m . ; B Ž r X , xX , « X . . Hence the homotopy H contracts the ball B Ž r X , xX , b . into a point in the ball B Ž r X , xX , « X .. Compactness of B Ž r X , xX , b . follows from Ž1., Ž4., and from the compactness of B Ž r , x, d . . Hence the number b lies in the intersection =Ž r X , xX , « X . l Ž d y s , d q s .. Theorem Ž1.2. is thus proved. LEMMA Ž3.1.. Let Z be a Hausdorff space and suppose that the function f : Z ª R* is locally positi¨ e. Then F : z ¬ Ž0, f Ž z .x is a quasi lower semicontinuous and closed-¨ alued mapping into the complete metric space R*. Proof. Local positivity of f means that for each z g Z, there exists a neighborhood UŽ z . such that inf f Ž zX . ¬ zX g UŽ z .4 ) 0. To check the quasi lower semicontinuity of F at a point z g Z it suffices to choose points q Ž z . in which the values of the function f approximate the above infimum. More precisely, for a fixed z g Z, fixed g ) 0, and a fixed neighborhood V Ž z . we put m s inf f Ž zX . ¬ zX g U Ž z . l V Ž z . 4 ) 0. If m s ` then F ' R* over the whole neighborhood W s UŽ z . l V Ž z . and in this case we can set q Ž z . s z. If m - ` then we choose a positive l such that cŽ m, m q l. - g and then we choose a point q Ž z . g W such that m - f Ž q Ž z .. - m q l.
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Let us check that the following implication holds: t g F Ž q Ž z . . « W ; Fy1 Ž B Ž c, t , g . . . For each point zX g W, we have that F Ž zX . s Ž0, m x j Ž m, f Ž zX .x and F Ž q Ž z .. s Ž0, m x j Ž m, f Ž q Ž z ..x. Hence only three cases can occur: Ža. t F m; Žb. f Ž zX . ) f Ž q Ž z ..; and Žc. the numbers t and f Ž zX . lie in the segment w m, f Ž q Ž z ..x. In the cases Ža. and Žb., t g F Ž zX . l B Ž c, t , g . / B, whereas in the case Žc., cŽ t, f Ž zX .. - cŽ m, m q l. - g and hence f Ž zX . g F Ž zX . l B Ž c, t , g . / B. Proof of Theorem Ž1.3.. By Lemma Ž3.1. it suffices to check the local positivity of the function d 0 : Ž r , x, « . ¬ sup DŽ r , x, « ., i.e., the property that for every z s Ž r , x, « . g Z s M X = X = Ž0,`., there exists a neighborhood UŽ z . such that inf d 0 Ž zX . ¬ zX g U Ž z . 4 ) 0. For a fixed point z g Z choose the number 2 d from the set DŽ r , x, «r2. and put a s min «r6, dr3, 14 . Let UŽ z . be the Cartesian product of the a-neighborhoods of points r g Ž M X , dist., x g Ž X, r ., « Ž0, `.. We claim that d g DŽ zX . for all zX g UŽ z ., i.e., that inf d 0 Ž zX . ¬ zX g U Ž z . 4 G d ) 0. In fact, let zX s Ž r X , xX , « X . and distŽ r , r X . - a F 1, r Ž x, xX . - a , < « y « X < - a . Then we can conclude from r Ž x, y . - «r2 that
r X Ž xX , y . F r Ž xX , y . q dist Ž r , r X . - r Ž xX , x . q r Ž x, y . q a - r Ž x, y . y 2 a - «r2 q 2 a F « y a - « X hence B Ž r , x, «r2 . ; B Ž r X , xX , « X . . X
Ž 6.
X
In an analogous manner we obtain from r Ž x , y . - d that
r Ž x, y . F r X Ž x, y . q dist Ž r , r X . - r X Ž x, xX . q r X Ž xX , y . q a - r X Ž xX , y . q a q r Ž x, xX . q dist Ž r , r X . - r X Ž xX , y . q 3 a - d q 3 a F 2 d hence B Ž r X , xX , d . ; B Ž r , x, 2 d . .
Ž 7.
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By our choice of the number d we have the ball B Ž r , x, 2 d . is contradictible to a point in the ball B Ž r , x, «r2. and in the ball B Ž r X , xX , « X ., because of Ž6.. By Ž7., the ball B Ž r X , xX , d . contracts into the same point Žunder the same homotopy. in the ball B Ž r X , xX , « X .. Hence d g DŽ zX ., for all zX g UŽ z .. Theorem Ž1.3. is thus proved. Proof of Theorem Ž1.5.. Quasi lower semicontinuity of the map D implies quasi lower semicontinuity of the map Dr2. Note that Dr2 is a selection of D. By Theorem Ž1.4., there exists a lower semicontinuous closed-value selection of the map Dr2, say F. By Michael’s selection theorem w8, Theorem 3.1Z x, the map conv F admits a singlevalued continuous selection. The last selection of the map conv F will be a selection of the map Dr2 and will be a selection of the modulus D of the local contractibility of the space X. Theorem Ž1.5. is thus also proved.
4. EPILOGUE We conclude with some remarks and observations: Ža. Let Ž X, d . and Ž Y, r . be metric spaces and let C Ž X, Y . be the space of all continuous mappings from X into Y, endowed with the uniform topology. Then for each triple Ž f, x, « . g C Ž X, Y . = X = Ž0, `. one can consider the set of all positive d such that d Ž x, xX . - d « r Ž f Ž x . , f Ž xX . . - « , i.e., one can define a multivalued mapping C Ž X , Y . = X = Ž 0, ` . ª Ž 0, ` . . In w11x, it was proved that for locally compact X such a mapping is always lower semicontinuous and hence admits a selection. However, in general, ‘‘modulus of continuity’’ is non-lower semicontinuous, i.e., the situation is the same as in the present paper. ŽSee also w7x, where the inequality r Ž f Ž x ., f Ž xX .. - « is replaced by some suitable predicate in variables f, x, xX , « and r .. Žb. In the proof of Lemma Ž3.1. above, the neighborhood W does not depend on the choice of t g F Ž q Ž z ... Hence we in fact, prove that in this lemma Žand hence also in Theorem Ž1.3.. the multivalued mapping is weakly Hausdorff lower semicontinuous Žfor definition see w1x.. But de Blasi and Myjak’s theorem w1x is not directly applicable due to the incompleteness of the values F Ž z . in R.
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Žc. Formally, Theorem Ž1.5. can be proved using our proof of Theorem Ž1.3., avoiding Theorem Ž1.4. and the notion of quasi lower semicontinuity. In fact, local positivity of the function d 0 implies the existence of a local continuous Žconstant. selection for a map D. Because of paracompactness of the space Z s M X = X = Ž0, `. we can obtain a global singlevalued selection for D: it suffices to use a suitable continuous partition of unity. Žd. We conclude by stating interesting open problems. It is known that every quasi lower semicontinuous mapping F admits some ‘‘maximal’’ lower semicontinuous selection F0 . Moreover, such a selection coincides with the derived Žin the sense of Brown w2x. mapping F X of the mapping F. Problem Ž4.1.. Under the assumptions of Theorem Ž1.2., is it true that = s DX ? Problem Ž4.2.. Is it true that =n is lower semicontinuous whenever X is locally compact?
REFERENCES 1. F. de Blasi and J. Myjak, Continuous selections for weakly Hausdorff lower semicontinuous multifunctions, Proc. Amer. Math. Soc. 93 Ž1985., 369]372. 2. A. L. Brown, Set-valued mappings, continuous selections and metric projections, J. Approx. Theory 57 Ž1989., 48]68. 3. F. Deutsch and P. Kenderov, Continous selections and approximative selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14 Ž1983., 185]194. 4. V. Gutev, Unified selection and factorization theorems, C. R. Acad. Bulgare Sci. 40, No. 5 Ž1987., 13]15. 5. V. Gutev, Selection theorems under an assumption weaker than lower semi-continuity, Topology Appl. 50 Ž1993., 129]138. 6. W. Lin, Continous selections for set valued mappings, J. Math. Anal. Appl. 188 Ž1994., 1067]1072. 7. J. Malesic ˇ ˇ and D. Repovs, ˇ Continuity-like properties and continuous selections, Acta Math. Hungar. 73 Ž1996. 141]154. 8. E. Michael, Continuous selections, I, Ann. of Math. (2) 63 Ž1956., 361]382. 9. E. Michael, Continuous selections, III, Ann. of Math. (2) 65 Ž1957., 374]381. 10. J. Przeslawski and L. Rybynski, Michael continuous selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109 Ž1990., 537]543. 11. D. Repovs ˇ and P. V. Semenov, On continuous choice in the definition of continuity, Glas. Mat. 29, No. 49 Ž1994. 169]173.
216, 86]93 Ž1997.
AY975660
On Continuity Properties of the Modulus of Local Contractibility Dusan ˇ Repovs* ˇ Institute of Mathematics, Physics, and Mechanics, Uni¨ ersity of Ljubljana, P.O. Box 2964, 1001, Ljubljana, Slo¨ enia
and Pavel V. Semenov † Moscow State Pedagogical Uni¨ ersity, Ul. M. Pyrogo¨ skaya 1, Moscow, 119882, Russia Submitted by Ying-ming Liu Received February 6, 1997
Let M X be the set of all metrics compatible with a given topology on a locally contractible space X and let for each triple z s Ž r , x, « . g M X = X = Ž0, `., DŽ z . be the set of all positive d such that the open d-neighborhood of x is contractible in the open «-neighborhood of x in metric r . We prove several continuity properties of the map D : M X = X = Ž0, `. ª Ž0, `. and then, using a selection theorem for non-lower semicontinuous mappings, show that D admits a continuous singlevalued selection. Similar, but somewhat different properties are also demonstrated for the modulus D n of local n-connectedness. Q 1997 Academic Press
1. INTRODUCTION A topological space X is said to be locally contractible if for every point x g X and for each of its neighborhoods U ; X there exists a neighborhood V ; X of x such that the inclusion V ; U is homotopically trivial. For a metric space X the notion of local contractibility admits a definition via the real-valued parameters, namely the radii of neighborhoods U and * Supported in part by the Ministry of Science and Technology of the Republic of Slovenia Grant J1-07039r94. E-mail address: [email protected]. † Supported in part by the International Science G. Soros Foundation Grant NFU000. 86 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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V. More precisely, let M X be the set of all metrics on X compatible with the given topology on X. The space M X is equipped with topology induced by the following metric of uniform convergence: dist Ž r , d . s sup min < r Ž x, y . y d Ž x, y . < , 1 4 < x, y g X 4 . For every triple Ž r , x, « . g M X = X = Ž0, `. we define DŽ r , x, « . as the set of all positive numbers d such that the d-neighborhood B Ž r , x, d . of the point x in metric r is contractible in the «-neighborhood B Ž r , x, « . of the point x in the same metric r . In this way, we define a multivalued mapping D : M X = X = Ž0, `. ª Ž0, `. with nonempty convex values which we call the modulus of local contractibility of the space X. The definition of modulus D n of local n-connectedness is very similar to the definition of modulus of local contractibility }one only needs to replace the contractibility of the ball B Ž r , x, d . inside the ball B Ž r , x, « . with the contractibility inside the ball B Ž r , x, « . of every continuous image of the n-sphere S n lying in B Ž r , x, d .. The goal of the present paper is to investigate some continuity type properties of D. As a corollary we prove the existence of a continuous singlevalued choice Žselection. of elements of the sets DŽ r , x, « . Žsee Theorem 1.5.. EXAMPLE Ž1.1.. Let X be the half-open interval w0, 1. and let r be the standard metric on R. Then DŽ r , 0, 1. s Ž0, `. and DŽ r , 0, « . s Ž0, « x, for all « - 1. This example shows that the map D does not need to be lower semicontinuous and hence in general, the standard Michael selection theory techniques do not apply. However, for a locally compact space X it is always possible to find a lower semicontinuous selection of the map D: THEOREM Ž1.2.. Let X be a locally contractible and locally compact metrizable space. Define the map = : M X = X = Ž0, `. ª Ž0, `. as follows: let for each triple Ž r , x, « . g M X = X = Ž0, `. = Ž r , x, « . s d g D Ž r , x, « . ¬ the closure of B Ž r , x, d . is compact 4 . Then the map = is lower semicontinuous. In general we can only prove the quasi lower semicontinuity of the closure of the modulus of local contractibility D. We denote by d 0 Ž r , x, « . s sup DŽ r , x, « . and DŽ r , x, « . s Ž0, d 0 Ž r , x, « .x. Clearly, DŽ r , x, « . is the closure of the set DŽ r , x, « . in the complete metric space R* s ŽŽ0, `. j `4 , c ., where c Ž t , s . s < ty1 y sy1 <
and
c Ž t , ` . s ty1 .
88
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THEOREM Ž1.3.. Let X be a locally contractible Ž locally n-connected. metrizable space. Then the map D : M X = X = Ž0, `. ª R* Ž respecti¨ ely, the map D n . is a quasi lower semicontinuous mapping into the complete metric space with closed, con¨ ex ¨ alues. THEOREM Ž1.4. ŽGutev w5x.. Any quasi lower semicontinuous closed ¨ alued mapping from a topological space into a complete metric space admits a lower semicontinuous closed-¨ alue selection. THEOREM Ž1.5.. Let X be a locally contractible Ž locally n-connected. metrizable space. Then there exists a single¨ alued continuous function dˆ : M X = X = Ž0, `. ª Ž0, `. such that for e¨ ery point Ž r , x, « . g M X = X = Ž0, `., the neighborhood B Ž r , x, dˆŽ r , x, « .. is contractible in the neighborhood B Ž r , x, « . Ž respecti¨ ely, e¨ ery continuous mapping of the n-sphere S n into B Ž r , x, dˆŽ r , x, « .. is null homotopic in B Ž r , x, « ... Having the values of the multivalued mapping D as convex subsets of R one can try to use the well-known Deutsch]Kenderov approach w3x. More precisely, it is natural to start by proving almost lower semicontinuity or 2-lower semicontinuity of D or D Žsee w3x for definitions.. Unfortunately, their selection theorems work only for compact convex-valued mappings. A theorem from w6x generalizes the Deutsch]Kenderov theorem, but only for closed convex-valued mappings into R. So, we chose instead the Gutev selection theorem w5, Theorem 1.4x.
2. PRELIMINARIES Recall that a multivalued map G : X ª Y is said to be a selection of a multivalued map F : X ª Y if GŽ x . ; F Ž x ., for every x g X. A multivalued map F : X ª Y is said to be lower semicontinuous if for every open subset V ; Y, the set Fy1 Ž V . s x g X ¬ F Ž x . l V / B4 is open in the space X. For a metric space Ž Y, r ., the lower semicontinuity of F at a point x can be reformulated as follows: for each positive g the following implication holds: y g F Ž x . « x g Int Ž Fy1 Ž B Ž r , y, g . . . . DEFINITION Ž2.1. w4, 5, 10x. A multivalued mapping F : X ª Y of a topological space X into a metric space Ž Y, r . is said to be quasi lower semicontinuous at a point x g X if for each positive g and for each neighborhood V Ž x . there exists a point q Ž x . g V Ž x . such that the follow-
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ing implication holds: y g F Ž q Ž x . . « x g Int Ž Fy1 Ž B Ž r , y, g . . . . Quasi lower semicontinuity of F : X ª Y means that F is quasi lower semicontinuous at every point x g X. Clearly, quasi lower semicontinuity follows from the lower semicontinuity: it suffices to put q Ž x . s x. However, the converse does not hold. Let us consider an example: EXAMPLE Ž2.2.. Let a function f : R ª R* be monotone and increasing. Then the map F : t ¬ Ž0, f Ž t .x is quasi lower semicontinuous. In this example it is easy to construct a lower semicontinuous selection of F in a straightforward manner: it suffices to put H Ž t . s Ž0, f Ž t y 0.x ; F Ž t .. We can apply the standard Michael selection theorem w8, Theorem 3.1Z x to the map H and find a singlevalued continuous selection for H and hence for F. In particular, such a selection exists for the modulus of the uniform local contractibility of a metric space Ž Y, r ..
3. PROOFS OF THEOREMS Proof of Theorem Ž1.2.. Let us check the lower semicontinuity of the map = at a point Ž r , x, « . g M X = X = Ž0, `.. To this end, we fix d g =Ž r , x, « . and some of its s-neighborhoods Ž d y s , d q s .. We need to find a neighborhood V of the point Ž r , x, « ., such that for all Ž r X , xX , « X . g V, the sets =Ž r X , xX , « X . intersect with the interval Ž d y s , d q s .. If d y s F 0, then the sets =Ž r X , xX , « X . intersect with the interval Ž d y s , d q s . for all triples Ž r X , xX , « X .. In the case when d y s ) 0 we choose numbers a and b so that
dys-b-a-d
Ž 1.
and we fix a homotopy H : B Ž r , x, d . = w 0, 1 x ª B Ž r , x, « . which contracts the neighborhood B Ž r , x, d . into a point. Let us consider the image H ŽB Ž r , x, a . = w0, 1x., where B denotes the closure of B. Because of the compactness of the set B Ž r , x, d . this image lies in the open ball B Ž r , x, « . with some ‘‘freedom.’’ More precisely, there exists a number 0 - l - « such that H Ž B Ž r , x, a . = w 0, 1 x . ; B Ž r , x, l . .
Ž 2.
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Fix a number m such that l - m - « and put r s min Ž a y b . r2, Ž m y l . r2, 1 4 .
Ž 3.
Let V1 be the r-neighborhood of the metric r in Ž M X , dist., and let V2 be the r-neighborhood of the point x in metric r and let V3 s Ž m , `. be the neighborhood of the number « . Then by the triangle inequality B Ž r X , xX , b . ; B Ž r , x, a .
Ž 4.
B Ž r , x, l . ; B Ž r X , xX , m .
Ž 5.
and
for all Ž r X , xX . g V1 = V2 . Hence for each triple Ž r X , xX , « X . g V s V1 = V2 = V3 , we have that Ž4 .
H Ž B Ž r X , xX , b . = w 0, 1 x . ; H Ž B Ž r , x, a . = w 0, 1 x . Ž2 .
Ž5 .
; B Ž r , x, l . ; B Ž r X , xX , m . ; B Ž r X , xX , « X . . Hence the homotopy H contracts the ball B Ž r X , xX , b . into a point in the ball B Ž r X , xX , « X .. Compactness of B Ž r X , xX , b . follows from Ž1., Ž4., and from the compactness of B Ž r , x, d . . Hence the number b lies in the intersection =Ž r X , xX , « X . l Ž d y s , d q s .. Theorem Ž1.2. is thus proved. LEMMA Ž3.1.. Let Z be a Hausdorff space and suppose that the function f : Z ª R* is locally positi¨ e. Then F : z ¬ Ž0, f Ž z .x is a quasi lower semicontinuous and closed-¨ alued mapping into the complete metric space R*. Proof. Local positivity of f means that for each z g Z, there exists a neighborhood UŽ z . such that inf f Ž zX . ¬ zX g UŽ z .4 ) 0. To check the quasi lower semicontinuity of F at a point z g Z it suffices to choose points q Ž z . in which the values of the function f approximate the above infimum. More precisely, for a fixed z g Z, fixed g ) 0, and a fixed neighborhood V Ž z . we put m s inf f Ž zX . ¬ zX g U Ž z . l V Ž z . 4 ) 0. If m s ` then F ' R* over the whole neighborhood W s UŽ z . l V Ž z . and in this case we can set q Ž z . s z. If m - ` then we choose a positive l such that cŽ m, m q l. - g and then we choose a point q Ž z . g W such that m - f Ž q Ž z .. - m q l.
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Let us check that the following implication holds: t g F Ž q Ž z . . « W ; Fy1 Ž B Ž c, t , g . . . For each point zX g W, we have that F Ž zX . s Ž0, m x j Ž m, f Ž zX .x and F Ž q Ž z .. s Ž0, m x j Ž m, f Ž q Ž z ..x. Hence only three cases can occur: Ža. t F m; Žb. f Ž zX . ) f Ž q Ž z ..; and Žc. the numbers t and f Ž zX . lie in the segment w m, f Ž q Ž z ..x. In the cases Ža. and Žb., t g F Ž zX . l B Ž c, t , g . / B, whereas in the case Žc., cŽ t, f Ž zX .. - cŽ m, m q l. - g and hence f Ž zX . g F Ž zX . l B Ž c, t , g . / B. Proof of Theorem Ž1.3.. By Lemma Ž3.1. it suffices to check the local positivity of the function d 0 : Ž r , x, « . ¬ sup DŽ r , x, « ., i.e., the property that for every z s Ž r , x, « . g Z s M X = X = Ž0,`., there exists a neighborhood UŽ z . such that inf d 0 Ž zX . ¬ zX g U Ž z . 4 ) 0. For a fixed point z g Z choose the number 2 d from the set DŽ r , x, «r2. and put a s min «r6, dr3, 14 . Let UŽ z . be the Cartesian product of the a-neighborhoods of points r g Ž M X , dist., x g Ž X, r ., « Ž0, `.. We claim that d g DŽ zX . for all zX g UŽ z ., i.e., that inf d 0 Ž zX . ¬ zX g U Ž z . 4 G d ) 0. In fact, let zX s Ž r X , xX , « X . and distŽ r , r X . - a F 1, r Ž x, xX . - a , < « y « X < - a . Then we can conclude from r Ž x, y . - «r2 that
r X Ž xX , y . F r Ž xX , y . q dist Ž r , r X . - r Ž xX , x . q r Ž x, y . q a - r Ž x, y . y 2 a - «r2 q 2 a F « y a - « X hence B Ž r , x, «r2 . ; B Ž r X , xX , « X . . X
Ž 6.
X
In an analogous manner we obtain from r Ž x , y . - d that
r Ž x, y . F r X Ž x, y . q dist Ž r , r X . - r X Ž x, xX . q r X Ž xX , y . q a - r X Ž xX , y . q a q r Ž x, xX . q dist Ž r , r X . - r X Ž xX , y . q 3 a - d q 3 a F 2 d hence B Ž r X , xX , d . ; B Ž r , x, 2 d . .
Ž 7.
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By our choice of the number d we have the ball B Ž r , x, 2 d . is contradictible to a point in the ball B Ž r , x, «r2. and in the ball B Ž r X , xX , « X ., because of Ž6.. By Ž7., the ball B Ž r X , xX , d . contracts into the same point Žunder the same homotopy. in the ball B Ž r X , xX , « X .. Hence d g DŽ zX ., for all zX g UŽ z .. Theorem Ž1.3. is thus proved. Proof of Theorem Ž1.5.. Quasi lower semicontinuity of the map D implies quasi lower semicontinuity of the map Dr2. Note that Dr2 is a selection of D. By Theorem Ž1.4., there exists a lower semicontinuous closed-value selection of the map Dr2, say F. By Michael’s selection theorem w8, Theorem 3.1Z x, the map conv F admits a singlevalued continuous selection. The last selection of the map conv F will be a selection of the map Dr2 and will be a selection of the modulus D of the local contractibility of the space X. Theorem Ž1.5. is thus also proved.
4. EPILOGUE We conclude with some remarks and observations: Ža. Let Ž X, d . and Ž Y, r . be metric spaces and let C Ž X, Y . be the space of all continuous mappings from X into Y, endowed with the uniform topology. Then for each triple Ž f, x, « . g C Ž X, Y . = X = Ž0, `. one can consider the set of all positive d such that d Ž x, xX . - d « r Ž f Ž x . , f Ž xX . . - « , i.e., one can define a multivalued mapping C Ž X , Y . = X = Ž 0, ` . ª Ž 0, ` . . In w11x, it was proved that for locally compact X such a mapping is always lower semicontinuous and hence admits a selection. However, in general, ‘‘modulus of continuity’’ is non-lower semicontinuous, i.e., the situation is the same as in the present paper. ŽSee also w7x, where the inequality r Ž f Ž x ., f Ž xX .. - « is replaced by some suitable predicate in variables f, x, xX , « and r .. Žb. In the proof of Lemma Ž3.1. above, the neighborhood W does not depend on the choice of t g F Ž q Ž z ... Hence we in fact, prove that in this lemma Žand hence also in Theorem Ž1.3.. the multivalued mapping is weakly Hausdorff lower semicontinuous Žfor definition see w1x.. But de Blasi and Myjak’s theorem w1x is not directly applicable due to the incompleteness of the values F Ž z . in R.
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Žc. Formally, Theorem Ž1.5. can be proved using our proof of Theorem Ž1.3., avoiding Theorem Ž1.4. and the notion of quasi lower semicontinuity. In fact, local positivity of the function d 0 implies the existence of a local continuous Žconstant. selection for a map D. Because of paracompactness of the space Z s M X = X = Ž0, `. we can obtain a global singlevalued selection for D: it suffices to use a suitable continuous partition of unity. Žd. We conclude by stating interesting open problems. It is known that every quasi lower semicontinuous mapping F admits some ‘‘maximal’’ lower semicontinuous selection F0 . Moreover, such a selection coincides with the derived Žin the sense of Brown w2x. mapping F X of the mapping F. Problem Ž4.1.. Under the assumptions of Theorem Ž1.2., is it true that = s DX ? Problem Ž4.2.. Is it true that =n is lower semicontinuous whenever X is locally compact?
REFERENCES 1. F. de Blasi and J. Myjak, Continuous selections for weakly Hausdorff lower semicontinuous multifunctions, Proc. Amer. Math. Soc. 93 Ž1985., 369]372. 2. A. L. Brown, Set-valued mappings, continuous selections and metric projections, J. Approx. Theory 57 Ž1989., 48]68. 3. F. Deutsch and P. Kenderov, Continous selections and approximative selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14 Ž1983., 185]194. 4. V. Gutev, Unified selection and factorization theorems, C. R. Acad. Bulgare Sci. 40, No. 5 Ž1987., 13]15. 5. V. Gutev, Selection theorems under an assumption weaker than lower semi-continuity, Topology Appl. 50 Ž1993., 129]138. 6. W. Lin, Continous selections for set valued mappings, J. Math. Anal. Appl. 188 Ž1994., 1067]1072. 7. J. Malesic ˇ ˇ and D. Repovs, ˇ Continuity-like properties and continuous selections, Acta Math. Hungar. 73 Ž1996. 141]154. 8. E. Michael, Continuous selections, I, Ann. of Math. (2) 63 Ž1956., 361]382. 9. E. Michael, Continuous selections, III, Ann. of Math. (2) 65 Ž1957., 374]381. 10. J. Przeslawski and L. Rybynski, Michael continuous selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109 Ž1990., 537]543. 11. D. Repovs ˇ and P. V. Semenov, On continuous choice in the definition of continuity, Glas. Mat. 29, No. 49 Ž1994. 169]173.