nLab
disjoint union 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

Contents
Definition
Given a set $X_i$ , $i \in I$ of topological space , then their disjoint union space $\underset{i \in I}{\sqcup} X_i$ is the topological space whose underlying set is the disjoint union of the underlying sets of the $X_i$ , and whose open subsets are precisely the disjoint unions of the open subsets of the $X_i$ .

More abstractly, this is the coproduct in the category Top of topological spaces.

examples of universal constructions of topological spaces :

$\phantom{AAAA}$ limits $\phantom{AAAA}$ colimits $\,$ point space $\,$ $\,$ empty space $\,$
$\,$ product topological space $\,$ $\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$ $\,$ quotient topological space $\,$
$\,$ fiber space $\,$ $\,$ space attachment $\,$
$\,$ mapping cocylinder , mapping cocone $\,$ $\,$ mapping cylinder , mapping cone , mapping telescope $\,$
$\,$ cell complex , CW-complex $\,$

Revised on May 14, 2017 07:23:57
by

Urs Schreiber
(92.218.150.85)