The general definition makes sense for arbitrary coverings and does not require a topology. Let ${\displaystyle X}$  be a set and let ${\displaystyle {\mathcal {U}}}$  be a covering of ${\displaystyle X,}$  that is, ${\textstyle X=\bigcup {\mathcal {U}}.}$  Given a subset ${\displaystyle S}$  of ${\displaystyle X,}$  the star of ${\displaystyle S}$  with respect to ${\displaystyle {\mathcal {U}}}$  is the union of all the sets ${\displaystyle U\in {\mathcal {U}}}$  that intersect ${\displaystyle S,}$  that is,

Given a point ${\displaystyle x\in X,}$  we write ${\displaystyle \operatorname {st} (x,{\mathcal {U}})}$  instead of ${\displaystyle \operatorname {st} (\{x\},{\mathcal {U}}).}$ 

A covering ${\displaystyle {\mathcal {U}}}$  of ${\displaystyle X}$  is a refinement of a covering ${\displaystyle {\mathcal {V}}}$  of ${\displaystyle X}$  if every ${\displaystyle U\in {\mathcal {U}}}$  is contained in some ${\displaystyle V\in {\mathcal {V}}.}$  The following are two special kinds of refinement. The covering ${\displaystyle {\mathcal {U}}}$  is called a barycentric refinement of ${\displaystyle {\mathcal {V}}}$  if for every ${\displaystyle x\in X}$  the star ${\displaystyle \operatorname {st} (x,{\mathcal {U}})}$  is contained in some ${\displaystyle V\in {\mathcal {V}}.}$ ^[1][2] The covering ${\displaystyle {\mathcal {U}}}$  is called a star refinement of ${\displaystyle {\mathcal {V}}}$  if for every ${\displaystyle U\in {\mathcal {U}}}$  the star ${\displaystyle \operatorname {st} (U,{\mathcal {U}})}$  is contained in some ${\displaystyle V\in {\mathcal {V}}.}$ ^[3][2]

Every star refinement of a cover is a barycentric refinement of that cover. The converse is not true, but a barycentric refinement of a barycentric refinement is a star refinement.^[4][5][6][7]

Given a metric space ${\displaystyle X,}$  let ${\displaystyle {\mathcal {V}}=\{B_{\epsilon }(x):x\in X\}}$  be the collection of all open balls ${\displaystyle B_{\epsilon }(x)}$  of a fixed radius ${\displaystyle \epsilon >0.}$  The collection ${\displaystyle {\mathcal {U}}=\{B_{\epsilon /2}(x):x\in X\}}$  is a barycentric refinement of ${\displaystyle {\mathcal {V}},}$  and the collection ${\displaystyle {\mathcal {W}}=\{B_{\epsilon /3}(x):x\in X\}}$  is a star refinement of ${\displaystyle {\mathcal {V}}.}$ 

• Family of sets – Any collection of sets, or subsets of a set

• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.