title: Point-finite collection
text: # Point-finite collection
  
  In mathematics, a collection or family ${\mathcal {U}}$ of subsets of a topological space $X$ is said to be point-finite if every point of $X$ lies in only finitely many members of ${\mathcal {U}}.$
  
  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.
  
  