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p-regular

Last-modified: 2012-10-04 (木) 14:33:34

Definition Edit

  • A topological space (X,τ) is said to be p-regular if for each closed set F of X and each point imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20X%5csetminus%20F%20%5c%5d%7d%25.png , there exist disjoint pre-open? sets U and V such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20F%5csubset%20U%20%5cmbox%7b%20and%20%7d%20x%5cin%20V%20%5c%5d%7d%25.png.

Property Edit

  • For a topological space (X,τ) the following are equivalent:
    1. X is p-regular.
    2. For each x in X and each open set U of X containing x, there exists pre-open? set V such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20V%5csubset%20p%5cmbox%7b-%7dcl%28V%29%5csubset%20U%20%5c%5d%7d%25.png .
    3. For each closed set F of X, imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5ccap%5c%7bp%5cmbox%7b-%7dcl%28V%29%5c%20%7c%5c%20V%5cmbox%7b%20is%20pre-open%20set%20such%20that%7d%5c%20F%5csubset%20V%5c%7d=F%20%5c%5d%7d%25.png
    4. For each subset A of X and each open set U of X such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20U%5cneq%5cemptyset%20%5c%5d%7d%25.png , there exists pre-openset? V such taht imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20V%5cneq%5cemptyset%5cmbox%7b%20and%20%7dp%5cmbox%7b-%7dcl%28V%29%5csubset%20U%20%5c%5d%7d%25.png .
    5. For each nonempty subset A of X and each closed set F of X such tant imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20F=%5cemptyset%20%5c%5d%7d%25.png , there exist pre-open? sets V and W such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20V%5cneq%5cemptyset%2c%5c%20F%5csubset%20W%5cmbox%7b%20and%20%7dV%5ccap%20W=%5cemptyset%20%5c%5d%7d%25.png .
  • The product space of p-regular spaces is p-regular.
  • If X is a p-regular space and X' is an α-set of X, then the subspace X' is p-regular.

Reference Edit

  • El-Deeb, N.; Hasanein, I. A.; Mashhour, A. S.; Noiri, T. , On p-regular spaces., Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 27(75) (1983), no. 4, 311–315.