Real Analysis Exchange

Classification of Points of Lower Semi-continuity of a Multifunction in Topological Spaces

Hiranmay Dasgupta and Saibal Ranjan Ghosh

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Abstract

In this paper we introduce the notion of $y$-lower semi-continuity and point out a distinction between a point of lower semi-continuity in global sense and a point of lower semi-continuity in local sense in general topological spaces after classifying points of $y$-lower semi-continuity (resp. lower semi-continuity) and also study their interrelationships. In particular, we find a necessary and sufficient condition for a bijective open multifunction on a $T_{2}$ space to be lower semi-continuous. Finally, a sufficient condition for an open bijective multifunction on the real line to have at most countable points of lower semi-discontinuity is formulated.

Article information

Source
Real Anal. Exchange, Volume 36, Number 1 (2010), 29-44.

Dates
First available in Project Euclid: 14 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.rae/1300108082

Mathematical Reviews number (MathSciNet)
MR3016401

Zentralblatt MATH identifier
1244.54033

Subjects
Primary: 54C05: Continuous maps 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
Secondary: 54A99: None of the above, but in this section

Keywords
lower semi-continuity lower semi-discontinuity \textity-l.s.c. \textity-l.s.d. $s_y$-point $w_y$-point honest point \textits-point \textitw-point \textitc-point

Citation

Ghosh, Saibal Ranjan; Dasgupta, Hiranmay. Classification of Points of Lower Semi-continuity of a Multifunction in Topological Spaces. Real Anal. Exchange 36 (2010), no. 1, 29--44. https://projecteuclid.org/euclid.rae/1300108082


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References

  • J. Cao and W. B. Moors, Quasicontinuous selections of upper continuous set-valued mappings, Real Anal. Exchange, 31(1) (2005), 1-7.
  • E. Ekici, On almost and weak forms of nearly continuous multifunctions, Proc. Jangjeon Math. Soc., 9(2) (2006), 109-120.
  • S. R. Ghosh and H. Dasgupta, Classification of points of continuities and discontinuities of a mapping in topological spaces, Bull. Calcutta Math. Soc., 97(4) (2005), 282-296.
  • R. E. Smithson, Multifunctions, Nieuw Arch. Wiskd. (5), XX (1972), 31-53.
  • R. Vaidyanathaswamy, Set topology, Chelsea Publishing Company, New York, 2nd Edition (1960).
  • G. T. Whyburn, Continuity of multifunctions, Proc. Natl. Acad. Sci. USA, 54 (1965), 1494-1501.