$T_0$ topological spaces and $T_0$ posets in the topos of M-sets

M.M. Ebrahimi, M. Mahmoudi, and A.H. Nejah

In this paper, we introduce the concept of a topological space in the topos M-Set of M-sets, for a monoid M. We do this by replacing the notion of open "subset" by open "subobject" in the definition of a topology. We prove that the resulting category has an open subobject classifier, which is the counterpart of the Sierpinski space in this topos. We also study the relation between the given notion of topology and the notion of a poset in this universe. In fact, the counterpart of the specialization pre-order is given for topological spaces in M-Set, and it is shown that, similar to the classic case, for a special kind of topological spaces in M-Set, namely $T_0$ ones, it is a partial order. Furthermore, we obtain the universal $T_0$ space, and give the adjunction between topological spaces and $T_0$ posets, in M-Set.

Keywords: Topos, M-set, M-topological space, M-poset, M-continuous map, $T_{0}$ M-topological space, $T_0$ M-poset

2010 MSC: 18A40, 18B25, 06A06, 06D22, 54D10, 20M30

Theory and Applications of Categories, Vol. 33, 2018, No. 34, pp 1059-1071.

Published 2018-11-01.


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