AN ALTERNATIVE THEOREM FOR CONTINUOUS RELATIONS AND ITS APPLICATIONS

Ákos Münnich and Árpád Száz

Abstract: 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 open-valued and $V$ is closed, (iii) $S^{-1}$ is open-valued and $V$ is open, (iv) both $S$ and $S^{-1}$ are open-valued; 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 closed-valued, lower or upper semicontinuous relations (multifunctions)

Classification (MSC2000): 54C60

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