sums of compact pavings are compact

Suppose that (Ki,𝒦i) is a paved space for each i in an index setMathworldPlanetmathPlanetmath I. The direct sum, or disjoint unionMathworldPlanetmath (, βˆ‘i∈IKi is the union of the disjoint sets KiΓ—{i}. The direct sum of the paving 𝒦i is defined as

βˆ‘i∈I𝒦i={βˆ‘i∈ISi:Siβˆˆπ’¦iβˆͺ{βˆ…}⁒ is empty for all but finitely many ⁒i}.

Let (Ki,Ki) be compactPlanetmathPlanetmath paved spaces for i∈I. Then, βˆ‘iKi is a compact paving on βˆ‘iKi.

The paving 𝒦′ consisting of subsets of βˆ‘i𝒦i of the form βˆ‘iSi where Si=βˆ… for all but a single i∈I is easily shown to be compact. Indeed, if π’¦β€²β€²βŠ†π’¦β€² satisfies the finite intersection property then there is an i∈I such that SβŠ†KiΓ—{i} for every Sβˆˆπ’¦β€²β€². Compactness of 𝒦i gives β‹‚π’¦β€²β€²β‰ βˆ….

Then, as βˆ‘i𝒦i consists of finite unions of sets in 𝒦′, it is a compact paving (see compact pavings are closed subsets of a compact space).

Title sums of compact pavings are compact
Canonical name SumsOfCompactPavingsAreCompact
Date of creation 2013-03-22 18:45:15
Last modified on 2013-03-22 18:45:15
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 28A05
Synonym disjoint unions of compact pavings are compact
Related topic ProductsOfCompactPavingsAreCompact
Defines direct sum of pavings
Defines disjoint union of pavings