PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 33(47), pp. 163168 (1983) 

AN ALTERNATIVE THEOREM FOR CONTINUOUS RELATIONS AND ITS APPLICATIONSÁkos Münnich and Árpád SzázMathematical Institute, Lajos Kossuth University H4010 Debrecen, HungaryAbstract: In this paper, improving [10, Lemma 3.5] of M. S. Stanojevi\'c, we prove the following alternative theorem: If $S$ is a continuous relation from a connected space $X$ into a space $Y$ and $V$ is a subset of $Y$ such that at least one of the following conditions is fulfilled: (i) $V$ is both open and closed, (ii) $S$ is openvalued and $V$ is closed, (iii) $S^{1}$ is openvalued and $V$ is open, (iv) both $S$ and $S^{1}$ are openvalued; then either $S(x)\subset V$ for all $x\in X$, or $S(x)\setminus V\neq \emptyset$ for all $x\in X$. Keywords: Open or closedvalued, lower or upper semicontinuous relations (multifunctions) Classification (MSC2000): 54C60 Full text of the article:
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