Porosity point

of a set in a metric space

A point for which there exists a sequence of open balls with radii and common centre at the point , such that for any there is an open ball with radius , where is positive and independent of (but, generally speaking, depends on and ). A set is called porous if any point in it is a porosity point of it. A set is called -porous if it can be represented as a finite or countable union of porous sets (see [1]). A porosity point of is a porosity point of its closure . If , a porosity point of a set is not a Lebesgue density point either of or of . Any porous or -porous set is of the first Baire category (cf. Baire classes) and of Lebesgue measure zero in . The converse of this statement, generally speaking, does not hold: There even exist nowhere-dense perfect sets of measure zero that are not -porous (see [2]).

Sometimes, in the case of an infinite-dimensional space , porous and -porous sets take the role of sets of measure zero. If and is an increasing continuous function with , then is called an -porous point of a set if , using the same notations ( is independent of ). The concepts of -porous and --porous sets are defined accordingly. In the case (), a -porous set may be a set of positive Lebesgue measure.

References

[1]	E.P. Dolzhenko, "Boundary properties of arbitrary functions" Math. USSR Izv. , 1 : 1 (1967) pp. 1–12 Izv. Akad. Nauk. SSSR Ser. Mat. , 31 : 1 (1967) pp. 3–14
[2]	L. Zajiček, "Sets of -porosity and sets of -porosity " Casopis. Pešt. Mat. , 101 (1976) pp. 350–359
[3]	J. Foran, P.D. Humke, "Some set-theoretic properties of -porous sets" Real Anal. Exch. , 6 : 1 (1980/81) pp. 114–119
[4]	J. Tkadlec, "Constructions of some non---porous sets on the real line" Real Anal. Exch. , 9 : 2 (1983/84) pp. 473–482
[5]	S.J. Agronsky, A.M. Bruckner, "Local compactness and porosity in metric spaces" Real Anal. Exch. , 11 : 2 (1985/86) pp. 365–379
[6]	Yu.A. Shevchenko, "On Vitali's covering theorem" Vestnik Moskov. Univ. Ser. 1. Mat. Mech. : 3 (1989) pp. 11–14 (In Russian)