nLab
effective topological space
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

Constructivism, Realizability, Computability
Contents
Idea
A kind of topological space about which one can reason “effectively”, hence constructively . Used in computable analysis , see at computable function (analysis) .

formal topology Every effective topological space $X$ defines a formal space. If $X$ is ‘constructively complete’, then the formal points of $X$ coincide with its effective points; see Spreen10 .
equilogical space
References
Dieter Spreen, On effective topological spaces , The Journal of Symbolic Logic, Vol. 63, No. 1, Mar., 1998 (JSTOR )

Dieter Spreen, Effectively Given Spaces, Domains, and Formal Topology, 2010 PDF

See the references at computable analysis .

Discussion in relation to equilogical spaces is in

With an eye twoards application in computable physics the definition also appears as def. 2.2 in

Klaus Weihrauch , Ning Zhong, Is wave propagation computable or can wave computers beat the Turing machine? , Proc. of the London Math. Soc. (3) 85 (2002) (web )
Last revised on March 9, 2014 at 09:38:01.
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