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 ![]() ![]() ![]() |
[3] | J. Foran, P.D. Humke, "Some set-theoretic properties of ![]() |
[4] | J. Tkadlec, "Constructions of some non-![]() ![]() |
[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) |
Porosity point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Porosity_point&oldid=48246