nLab specialization order

The specialisation order

Context

Topology

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

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

(0,1)(0,1)-Category theory

The specialisation order

Idea

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

Definition

Definition

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.)

Properties

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 January 31, 2023 at 18:59:19. See the history of this page for a list of all contributions to it.