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

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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.

http://www.tac.mta.ca/tac/volumes/33/34/33-34.pdf

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