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--regular

Last-modified: 2011-08-21 () 16:11:41

Definition Edit

  • A topological space (X,) with an operation on is called --regular if for each
    --closed set F of X not containing x, there exist disjoint
    --open sets U and V such that x U and F V .
 

Property Edit

  • The following are equivalent for a topological space (X,) with an operation on :
    1. X is --regular.
    2. For each x X and each U -O(X, x), there exists a V -O(X, x) such that x V -cl(V) U.
    3. For each --closed set F of X, {- cl(V) : F V, V -O(X)} = F.
    4. For each A subset of X and each U -O(X) with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20U%5cneq%20%5cemptyset%20%5c%5d%7d%25.png, there exists a V -O(X) such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20V%5cneq%20%5cemptyset%20%5c%5d%7d%25.png and -cl(V) U.
    5. For each nonempty subset A of X and each --closed subset F of X with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20F=%20%5cemptyset%20%5c%5d%7d%25.png, there exists V,W -O(X) such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20V%5cneq%20%5cemptyset%20%5c%5d%7d%25.png, F W and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20W%5ccap%20V=%20%5cemptyset%20%5c%5d%7d%25.png.
    6. For each - -closed set F and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cnotin%20F%20%5c%5d%7d%25.png, there exists U -O(X) and a -g-open set V such that x U, F V and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20U%5ccap%20V=%20%5cemptyset%20%5c%5d%7d%25.png.
    7. For each A X and each --closed set F with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20F=%20%5cemptyset%20%5c%5d%7d%25.png, there exists U -O(X) and a -g-open set V such thatimgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20U%5cneq%20%5cemptyset%20%5c%5d%7d%25.png, F V and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20U%5ccap%20V=%20%5cemptyset%20%5c%5d%7d%25.png.
    8. For each --closed set F of X, F ={-cl(V) : F V, V is -g-open}.
 

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.