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--normal

Last-modified: 2011-08-21 () 16:10:07

Definition Edit

  • A topological space (X,) with an operation on is said to be --
    normal if for any pair of disjoint --closed sets A, B of X, there exist disjoint --open sets U and V such that A U and B V .
 

Property Edit

For a topological space (X,) with an operation on , the following are equivalent:

  1. X is --normal.
  2. For each pair of disjoint --closed sets A, B of X, there exist disjoint -g-open
    sets U and V such that A U and B V .
  3. For each --closed A and any --open set V containing A, there exists a-g-open set U such that A U -cl(U) V .
  4. For each --closed set A and any -g-open set B containing A, there exists a -g-open set U such that A U -cl(U) -int(B).
  5. For each --closed set A and any -g-open set B containing A, there exists a --open set G such that A G -cl(G) -int(B).
  6. For each -g-closed set A and any --open set B containing A, there exists a --open set U such that -cl(A) U -cl(U) B.
  7. For each -g-closed set A and any --open set B containing A, there exists a -g-open set G such that -cl(A) G -cl (G) B.
 

Reference Edit

  • Basu, C. K.; Afsan, B. M. Uzzal; Ghosh, M. K., A class of functions and separation axioms with respect to an operation. (English summary), Hacet. J. Math. Stat. 38 (2009), no. 2, 103-118.