nLab specialization order

The specialisation order



topology (point-set topology, point-free topology)

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The specialisation order


The specialisation order is a way of turning any topological space XX into a preordered set (with the same underlying set).



Given a topological space XX with topology π’ͺ(X)\mathcal{O}(X), the specialization order ≀\leq is defined by either of the following two equivalent conditions:

  1. x≀yx \leq y if and only if xx belongs to the topological closure of {y}\{y\}; we say that xx is a specialisation of yy.

  2. x≀yx \leq y if and only if βˆ€ U:π’ͺ(X)(x∈U)β‡’(y∈U).\forall_{U \colon \mathcal{O}(X)} (x \in U) \Rightarrow (y \in U).

(Note: some authors use the opposite ordering convention.)


The specialization order on a topological space is always a preorder, because the specialization order is defined as βˆ€ U:π’ͺ(X)(x∈U)β‡’(y∈U)\forall_{U \colon \mathcal{O}(X)} (x \in U) \Rightarrow (y \in U). Given any two sets AA and BB and any binary relation R(x,y)R(x, y) between AA and BB, the relation βˆ€ w:BR(x,w)β‡’R(y,w)\forall_{w \colon B} R(x, w) \implies R(y, w) is a preorder on AA.

XX is T 0T_0 if and only if its specialisation order is a partial order. XX is T 1T_1 iff its specialisation order is equality. XX is R 0R_0 (like T 1T_1 but without T 0T_0) iff its specialisation order is an equivalence relation. (See separation axioms.)

Given a continuous map f:X→Yf: X \to Y between topological spaces, it is order-preserving relative to the specialisation order. Thus, we have a faithful functor SpecSpec from the category of Top\Top of topological spaces to the category ProSet\ProSet of preordered sets.

In the other direction, to each proset XX we may associate a topological space whose elements are those of XX, and whose open sets are precisely the upward-closed sets with respect to the preorder. This topology is called the specialization topology. This defines a functor

i:ProSet→Topi \colon ProSet \to Top

which is a full embedding; the essential image of this functor is the category of Alexandroff spaces (spaces in which an arbitrary intersection of open sets is open). Hence the category of prosets is equivalent to the category of Alexandroff spaces.

In fact, we have an adjunction i⊣Speci \dashv Spec, making ProSetProSet a coreflective subcategory of TopTop. In particular, the counit evaluated at a space XX,

i(Spec(X))β†’X,i(Spec(X)) \to X,

is the identity function at the level of sets, and is continuous because any open UU of XX is upward-closed with respect to ≀\leq, according to the second equivalent condition of the definition of the specialization order.

This adjunction restricts to an adjoint equivalence between the categories FinPros\Fin\Pros and FinTop\Fin\Top of finite prosets and finite topological spaces. The unit and counit are both identity functions at the level of sets, so we in fact have an equivalence between these categories as concrete categories.

Last revised on June 6, 2023 at 08:59:05. See the history of this page for a list of all contributions to it.