p-regular - sepaxiom

*Definition [#gf6f626b]
-A topological space (X,τ) is said to be p-regular if for each closed set F of X and each point &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 [#t564300a]
-For a topological space (X,τ) the following  are equivalent:
++X is p-regular.
++For each x in X and each  open set U of X containing x, there exists [[pre-open]] set V such that &ref(http://www.eaflux.com/imgtex/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); .
++For each  closed set F of X, &ref(http://www.eaflux.com/imgtex/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);
++For each subset A of X and each open set U of X such that &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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); .
++For each nonempty subset A of X and each closed set F of X such tant &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 [#yee4e190]
-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&#8211;315.