# sums of compact pavings are compact

Suppose that $({K}_{i},{\mathrm{\pi \x9d\x92\xa6}}_{i})$ is a paved space for each $i$ in an index set^{} $I$. The direct sum, or disjoint union^{} (http://planetmath.org/DisjointUnion), ${\beta \x88\x91}_{i\beta \x88\x88I}{K}_{i}$ is the union of the disjoint sets ${K}_{i}\Gamma \x97\{i\}$. The direct sum of the paving ${\mathrm{\pi \x9d\x92\xa6}}_{i}$ is defined as

$$\underset{i\beta \x88\x88I}{\beta \x88\x91}{\mathrm{\pi \x9d\x92\xa6}}_{i}=\{\underset{i\beta \x88\x88I}{\beta \x88\x91}{S}_{i}:{S}_{i}\beta \x88\x88{\mathrm{\pi \x9d\x92\xa6}}_{i}\beta \x88\u037a\{\mathrm{\beta \x88\x85}\}\beta \x81\u2019\text{\Beta is empty for all but finitely many\Beta}\beta \x81\u2019i\}.$$ |

###### Theorem.

Let $\mathrm{(}{K}_{i}\mathrm{,}{\mathrm{K}}_{i}\mathrm{)}$ be compact^{} paved spaces for $i\mathrm{\beta \x88\x88}I$. Then, ${\mathrm{\beta \x88\x91}}_{i}{\mathrm{K}}_{i}$ is a compact paving on ${\mathrm{\beta \x88\x91}}_{i}{K}_{i}$.

The paving ${\mathrm{\pi \x9d\x92\xa6}}^{\beta \x80\xb2}$ consisting of subsets of ${\beta \x88\x91}_{i}{\mathrm{\pi \x9d\x92\xa6}}_{i}$ of the form ${\beta \x88\x91}_{i}{S}_{i}$ where ${S}_{i}=\mathrm{\beta \x88\x85}$ for all but a single $i\beta \x88\x88I$ is easily shown to be compact. Indeed, if ${\mathrm{\pi \x9d\x92\xa6}}^{\beta \x80\xb2\beta \x80\xb2}\beta \x8a\x86{\mathrm{\pi \x9d\x92\xa6}}^{\beta \x80\xb2}$ satisfies the finite intersection property then there is an $i\beta \x88\x88I$ such that $S\beta \x8a\x86{K}_{i}\Gamma \x97\{i\}$ for every $S\beta \x88\x88{\mathrm{\pi \x9d\x92\xa6}}^{\beta \x80\xb2\beta \x80\xb2}$. Compactness of ${\mathrm{\pi \x9d\x92\xa6}}_{i}$ gives $\beta \x8b\x82{\mathrm{\pi \x9d\x92\xa6}}^{\beta \x80\xb2\beta \x80\xb2}\beta \x89\mathrm{\beta \x88\x85}$.

Then, as ${\beta \x88\x91}_{i}{\mathrm{\pi \x9d\x92\xa6}}_{i}$ consists of finite unions of sets in ${\mathrm{\pi \x9d\x92\xa6}}^{\beta \x80\xb2}$, 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 |