Point-finite collection

Short description: Cover of a set

In mathematics, a collection or family [math]\displaystyle{ \mathcal{U} }[/math] of subsets of a topological space [math]\displaystyle{ X }[/math] is said to be point-finite if every point of [math]\displaystyle{ X }[/math] lies in only finitely many members of [math]\displaystyle{ \mathcal{U}. }[/math]

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.

References

• Willard, Stephen (2004). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. 
• Willard, Stephen (2012). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.