nLab disjoint union topological space

Definition

Given a set $X_i$, $i \in I$ of topological spaces, 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.

Related concepts

• disjoint union of spaces is used to characterize connected 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 $\,$