Point-finite collection
cover of a set
In mathematics, a collection or family
U
{\displaystyle {\mathcal {U}}}
of subsets of a topological space
X
{\displaystyle X}
is said to be point-finite if every point of
X
{\displaystyle X}
lies in only finitely many members of
U
.
{\displaystyle {\mathcal {U}}.}
A metacompact space is a topological space in which every open cover admits a point-finite open refinement. Every locally finite collection of subsets of a topological space is also point-finite.
A topological space in which every open cover admits a locally finite open refinement is called a paracompact space. Every paracompact space is therefore metacompact.
Dieudonné's theorem
The original proof uses Zorn's lemma, while Willard uses transfinite recursion.
References
This article incorporates material from point finite on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
This brief starts where responsible research should: with the source description of “Point-finite collection” as cover of a set. Everything that follows is an evidence route, not borrowed authority.
Why this record matters
The subject matters to the general reference register because the source frames it as cover of a set. Its deeper value depends on whether names, dates, institutions and citations support that framing.
Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Apr 22, 2026. The linked authority identifier is Q7207903. None of the 0 selected statements returned an explicit reference.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Point-finite collection”, its source revision and the description used here.
- Expand the search: follow Point-finite collection primary sources, Point-finite collection archive and Point-finite research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Point-finite collection”?
- What terminology or title could unlock a more precise catalogue search?
- Which cited source is closest to the event, object or claim?
Search terms from this dossier
This entry incorporates text from “Point-finite collection” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.