nLab
Scott topology
Context
Topology
topology (point-set topology , point-free topology )

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

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
Definition
The Scott topology on a preordered set is the topology in which the open subsets (called Scott-open ) are precisely those whose characteristic functions (from the given preorder into the preorder of truth values ) preserve directed joins (and this makes them necessarily monotonic ).

This in fact ensures that, in general, the continuous functions between preorders with Scott topologies are precisely those (necessarily monotonic) functions between them which preserve directed joins (called Scott-continuous ). The poset of truth values itself, therefore, when equipped with the Scott topology, becomes the open-set classifier , Sierpinski space .

Properties
As injective objects in $T_0$ -spaces
In the category of $T_0$ topological spaces (see at separation axiom ), the injective objects are precisely those given by Scott topologies on continuous lattices ; as locales these are locally compact and spatial .

Last revised on March 13, 2018 at 11:47:36.
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