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

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Given a set X iX_i, iIi \in I of topological space, then their disjoint union space iIX i\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 iX_i, and whose open subsets are precisely the disjoint unions of the open subsets of the X iX_i.

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

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\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)