Page 237 - Textos de Matemática Vol. 46

FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 227
Proof. Straightforward verification. ⇤
In the next section we describe the fibrous preorders arising from a topo- logical space, called spatial fibrous preorders and prove that the category of topological spaces is isomorphic to the category of spatial fibrous preorders. Before that we illustrate how the classical result of [1] can be obtained via this new setting.
Proposition 2.8. Let (B,⌧) be a topological space and consider the fibrous preorder, say G(B,⌧), described in Proposition 2.7. It is an Alexandrov space if and only if G(B,⌧) is isomorphic to a preorder.
Proof. If (B,⌧) is an Alexandrov space then there exists a map u: B ! A
assigning to each point x 2 B the element (✓x, x) 2 A with ✓x the intersection of all open neighbourhoods of x, moreover this map satisfies the requirements of Proposition 2.6, and hence it is isomorphic to a preorder.
For the converse, assume that G(B,⌧) is isomorphic to a preorder. Then u: B ! A provides, for each b 2 B, a smallest neighbourhood. Hence, (B,⌧) is
an Alexandrov space.
⇤
3. The main result
The so called spatial frames are the frames that are isomorphic to the topol- ogy of some space (see e.g. [10]). Here our main result is the description of the full subcategory of FibPreord0 which is equivalent to the category of topological spaces.
Definition 3.1. A fibrous preorder ( R @ // A p // B ) is said to be spatial when there exists s : B ! A and m : A ⇥B A ! A with A ⇥B A = {(a, a0 ) 2 A ⇥ A | p(a) = p(a0)} such that
(F4) ps(y) = y;
(F5) pm(a, a0) = p(a) = p(a0); (F6) m(a, a0)Ry ) aRy&a0Ry;
foreverya,a0 2Aandy2Bwithp(a)=p(a0).
Observe that the identification of fibrous morphisms, as in Definition 2.3, implies that the extra structure, (s, m), in the definition of a spatial fibrous pre- order, (R, A, B, @, p, s, m), is uniquely determined by its fibrous preorder part, that is (R, A, B, @, p). This means that the category of spatial fibrous preorders and (equivalent) fibrous morphisms is a full subcategory of FibPreord0, which we denote by SpFibPreord0.