Download PDF

star refinement∗
CWoo†
2013-03-21 22:06:48
Let X be a set and C = {Ci | i ∈ I} be a cover of X (we assume Ci and X
are all subsets of some universe). Let A ⊆ X. The star of A (with respect to
the cover C ) is defined as
[
?(A, C ) := {Ci ∈ C | Ci ∩ A 6= ∅}.
When A is a singleton, we write ?(x, C ) = ?({x}, C ).
Properties of ?
1. A ⊆ ?(A, C ).
2. If A ⊆ B, then ?(A, C ) ⊆ ?(B, C ).
3. For any cover C of X, the sets C ? := {?(Ci , C ) | Ci ∈ C } and C b :=
{?(x, C ) | x ∈ X} are both covers of X.
4. C C b C ? ( denotes cover refinement).
Definitions. Let C , D be two covers of X. If C ? D, then we say that C
is a star refinement of D, denoted by C ? D. If C b D, then we say that C
is a barycentric refinement of D, denoted by C b D.
Remark. By property 4 above, it is easy to see that C ? D ⇒ C b D ⇒
C D.
References
[1] S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
∗ hStarRefinementi
created: h2013-03-21i by: hCWooi version: h38959i Privacy setting:
h1i hDefinitioni h54A99i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1