Geodesic
An Alternative Theorem for Continuous Relations and its Applications
Publications de l'Institut Mathématique, _N_S_33 (1983) no. 47, p. 163 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

In this paper, improving [10, Lemma 3.5] of M. S. Stanojević, 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 $x\in X$.
Classification : 54C60
Keywords: Open or closed-valued, lower or upper semicontinuous relations (multifunctions) 
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     author = {\'Akos M\"unnich and \'Arp\'ad Sz\'az},
     title = {An {Alternative} {Theorem} for {Continuous} {Relations} and its {Applications}},
     journal = {Publications de l'Institut Math\'ematique},
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     publisher = {mathdoc},
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     number = {47},
     year = {1983},
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Ákos Münnich; Árpád Száz. An Alternative Theorem for Continuous Relations and its Applications. Publications de l'Institut Mathématique, _N_S_33 (1983) no. 47, p. 163 . http://geodesic.mathdoc.fr/item/PIM_1983_N_S_33_47_a22/