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 $A'=\coprod _{p,q\in P}A_{c}(q,p)$ 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 $\left\{X_{\alpha }\right\}_{\alpha \in A}$, we define the disjoint union topology on $\textstyle \coprod _{\alpha \in A}X_{\alpha }$ by declaring a subset of $\textstyle \coprod _{\alpha \in A}X_{\alpha }$ to be open if and only if its intersection with each $X_{\alpha }$ is open in $X_{\alpha }$.

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

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