disjoint union topology

disjoint union topology (plural disjoint union topologies)

1. (topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union.
   • 1978, Maria Louise Shea Terrell, Topological 2-categories and Principal Topological Categories, University of Virginia, page 19:

     The reader will recall that the morphism and object sets of ${\displaystyle A'=\coprod _{p,q\in P}A_{c}(q,p)}$  were not necessarily given the disjoint union topologies.

   • 2003, John M. Lee, Introduction to Smooth Manifolds, Springer, page 548:

     Given any indexed collection of topological spaces ${\displaystyle \left\{X_{\alpha }\right\}_{\alpha \in A}}$ , we define the disjoint union topology on ${\displaystyle \textstyle \coprod _{\alpha \in A}X_{\alpha }}$  by declaring a subset of ${\displaystyle \textstyle \coprod _{\alpha \in A}X_{\alpha }}$  to be open if and only if its intersection with each ${\displaystyle X_{\alpha }}$  is open in ${\displaystyle X_{\alpha }}$ .

   • 2005, Friedrich Ischebeck, Ravi A. Rao, Ideals and Reality: Projective Modules and Number of Generators of Ideals, Springer, page 118:

     And the set ${\displaystyle G^{n}(\mathbb {F} )}$  of all sub vector spaces of ${\displaystyle \mathbb {F} ^{n}}$ , i.e. the disjoint union of the ${\displaystyle G_{k}^{n}(\mathbb {F} )}$ , gets the disjoint union topology.

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