Page 231 - Textos de Matemática Vol. 46

FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS
NELSON MARTINS-FERREIRA
Dedicated to Manuela Sobral
Abstract. The well-known equivalence between preorders and Alexan- drov spaces is extended to an equivalence between arbitrary topological spaces and spatial fibrous preorders, a new notion to be introduced.
1. Introduction
The categorical equivalence between preorders and A-spaces (i.e. Alexandrov spaces) is essentially a classical result established in [1]. A preorder is simply a reflexive and transitive relation while an A-space is a topological space in which any intersection of open sets is open. The latter property trivially holds for fi- nite topological spaces and the equivalence between finite topological spaces and finite preorders was used in [6, 7] to solve some open problems in topolog- ical descent theory. In [4], p. 61, Ern´e writes “Hence the question arises: How can we enlarge the category of A-spaces on the one hand and the category of quasiordered sets on the other hand, so that we still keep an equivalence be- tween the topological and the order-theoretical structures, but many interesting ‘classical’ topologies are included in the extended definition?” and proposes the notions of B-space and C-space.
With a di↵erent motivation, and not being restricted to the order-theore- tical structures, we propose a new structure, which we call fibrous preorder, and which generalizes the one of a preorder. With the appropriate morphisms, called fibrous morphisms, and a suitable equivalence between them, we observe
Received: 28 July 2014 / Accepted: 10 January 2015.
2010 Mathematics Subject Classification. 18B35, 54H99, 16B50.
Key words and phrases. Preorder, quasiorder, topological space, fibrous preorder, fibrous
morphism, spatial fibrous preorder, equivalence of categories.
Research supported by IPLeiria(ESTG/CDRSP) and FCT grant SFRH/BPD/43216/
2008. Also by the FCT projects: PTDC/EME-CRO/120585/2010, PTDC/MAT/120222/ 2010 and PEst-OE/EME/UI4044/2013.
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