An Extension of Kelley's Closed Relation Theorem to Relator Spaces
by
Árpád Száz
Lajos Kossuth University, Debrecen

In [] we have proved the following straightforward extension of the dual of Lemma 36 of Kelley [] (p. 202).

Theorem 1 If \0( F ,  G )  is an almost uniformly lower semicontinuous closed pair of relations on a topologically semisymmetric relator space  \0X (\0\Cal R\0)  to a sequentially convergence-adherence complete metric type relator space  \0Y (\0\Cal S\0)  such that G Ì F ,  then

cl\0\CalR\0 æ è  G\0-1\0 æ è  V (\0y\0)  ö ø   ö ø \0  Ì \0 F\0-1\0 æ è  W  æ è  V (\0y\0)  ö ø   ö ø

for all   V ,  W\0 Î \0\Cal S  and  y Î Y\0.  Thus, in particular, the relation  F  and the pair  \0( G ,  F )  are uniformly lower semicontinuous.

Combining the  F=G  particular case of this theorem with those of Theorems 7.3 and 8.3  of  [, , #1#2]  we can at once get the next practically important

Theorem 2 If  F  is a relation on a reflexive, topologically semisymmetric relator space  \0X (\0\Cal R\0)  to a sequentially convergence-adherence complete metric type relator space  \0Y (\0\CalS\0) ,  then the following assertions are equivalent :

(1)   F  is closed and almost uniformly lower semicontinuous ;

(2)   F  is closed-valued and uniformly lower semicontinuous .

Remark The importance of this theorem lies mainly in the fact that if  F  is a linear relation on one vector relator space  \0X (\0\Cal R )  to another  \0Y (\0\Cal S\0)  such that the domain  \0F^\0-1\0(\0Y\0)  is not meager, then  F  is almost uniformly lower semicontinuous and  \0F^\0-1\0(\0Y\0)  is dense.

Acknowledgement The author's research has been supported by the grants OTKA T-016846 and FKFP 0310/1997.

1 L. Holá and I. Kupka Closed graph and open mapping theorems for linear relations Acta Math. Univ. Commenian. 46-47 1985 157-162 2 J\. L. Kelley General Topology Van Nostrand Reinhold New York 1955 3 J. Mala and Á. Száz Modifications of relators Acta Math. Hungar. 77 1997 69-81 4 Cs. Rakaczki and Á. Száz Semicontinuity and closedness properties of relations in relator spaces Techn. Rep. (Inst. Math. Univ. Debrecen) 97/16 23 pp 5 Á. Száz Structures derivable from relators Singularité 3/(8) 1992 14-30 6 Á. Száz Refinements of relators Techn. Rep. (Inst. Math. Univ. Debrecen) 93/76 19 pp 7 Á. Száz An extension of Kelley's closed relation theorem to relator spaces Techn. Rep. (Inst. Math. Univ. Debrecen) 97/17 23 pp 8 C. Ursescu Multifunctions with convex closed graph Czechoslovak Math. J. 25 1975 438-411 9 M. Wilhelm Criteria of openness of relations Fund. Math. 114 1981 219-228

Date received: May 8, 1998