disjoint union topology

disjoint union topology

English

Noun

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.

Translations

topology applicable to the disjoint union of a given set of topological spaces

Further reading

• Disjoint union (topology) on Wikipedia.Wikipedia