Slobodnik property
Recall that a Baire space is a topological space in which every non-empty open subset is of the second category in itself (cf. also Category of a set). A space is Baire if and only if the intersection of each countable family of dense open sets in
is dense (cf. also Dense set).
In what follows, consider a space equipped with two topologies,
and
, and assume that
is finer than
. A topology
has the Slobodnik property if the intersection of each countable family of
-open
-dense sets in
is
-dense. If
has the Slobodnik property, then
is a Baire space. Following a definition of A.R. Todd from [a3], the topologies
and
are
-related if for any subset
of
,
(the interior of
with respect to the topology
) is non-empty if and only if
is non-empty. If the topologies
and
are
-related, then
has the Slobodnik property if and only if
is a Baire space, and this is the case if and only if
is a Baire space.
Let be a function on
of the first Baire class in the topology
, i.e.
is a pointwise limit of a sequence of
-continuous functions (cf. also Baire classes). A very general problem emerges: How large can the set of all
-continuity points of
be? If
has the Slobodnik property, then
is
-continuous at all points of
except at a set of
-first category. This theorem generalizes Slobodnik's theorem from [a2]: Any limit of a sequence of separately continuous functions on the Euclidean space
is continuous on
, except at a set of the first category. Notice that separately continuous functions are of the first Baire class on
, that a function
is separately continuous on
exactly when it is continuous in the finer crosswise topology on
(a set
is open in this topology if for any
there is a
such that the "cross"
![]() |
is a subset of ), and that the crosswise topology has the Slobodnik property.
A more detailed investigation of the Slobodnik property and related notions can be found in [a1].
References
[a1] | J. Lukeš, J. Malý, L. Zajíček, "Fine topology methods in real analysis and potential theory" , Lecture Notes in Mathematics , 1189 , Springer (1986) |
[a2] | S.G. Slobodnik, "Expanding system of linearly closed sets" Mat. Zametki , 19 (1976) pp. 61–84 (In Russian) |
[a3] | A.R. Todd, "Quasiregular, pseudocomplete, and Baire spaces" Pacific J. Math. , 95 (1981) pp. 233–250 |
Slobodnik property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Slobodnik_property&oldid=12020