Page 243 - Textos de Matemática Vol. 46

FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 233
The p-adic topology. The p-adic topology on the set of integers is obtained as
N(n,x)={z2Z|z=x+kpn, k2Z}
If instead of a map N : N ⇥ B ! P(B) we consider a family of binary rela-
with B = Z.
tions Rn over B, then we have examples of the following type, with I = N.
Indexed families of preorders. A more general example is obtained as fol- lows. Let I be a unitary magma, B a set, (Ri)i2I a family of binary relations Ri ✓B⇥B,and(@i:Ri !I)afamilyofmapssuchthat:
(1) xRix
(2) xRijy ) xRiy&xRjy (3) xRib&bR@i(x,b)y ) xRiy
foralli,j2I andx,y,b2B.
A morphism, say from (@i : Ri ! I)i2I to (@i00 : Ri00 ! I0)i02I0 , consists of a
map f : B ! B0 together with a family of maps (fj : B ! I0)j2I0 such that xRfj(x)y ) f(x)Rj0 f(y).
In this case we construct a fibrous preorder as follows: A = I ⇥B, p(i, x) = x, s(x) = (1, x), m(i, j, x) = (ij, x),
(i, x)Ry , xRiy
and @(i, x, y) = (@i(x, y), y) if xRiy.
In the paper [5], in preparation, the topological spaces arriving from a struc-
ture such as the one above, which is called an abstract system of neighbourhoods, are considered and some of its categorical properties are studied.
5. Conclusion
In this note we have introduced the notions of (spatial) fibrous preorder and fibrous morphism, showing that the category of topological spaces is the quotient category of the category of spatial fibrous preorders, obtained by iden- tifying two fibrous morphisms whenever they have the same underlying map. The examples show that this notion provides a convenient setting to work with the intuitive notion of base of open neighbourhoods. However, as explained in the introduction, the main motivation that leads to the definition of fibrous preorder is the purpose of finding a purely categorical notion of topological space. In a future work we plan to specify the internal version of a fibrous pre- order, by replacing the relation R ✓ A ⇥ B with a jointly monomorphic pair of morphisms and by giving the appropriate translation of axioms (F1)-(F3)