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},
     pages = {163 },
     publisher = {mathdoc},
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     number = {47},
     year = {1983},
     language = {en},
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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/