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 $X$ is Baire if and only if the intersection of each countable family of dense open sets in $X$ is dense (cf. also Dense set).

In what follows, consider a space $X$ equipped with two topologies, $\rho$ and $\tau$, and assume that $\tau$ is finer than $\rho$. A topology $\tau$ has the Slobodnik property if the intersection of each countable family of $\tau$-open $\rho$-dense sets in $X$ is $\rho$-dense. If $\tau$ has the Slobodnik property, then $( X , \rho )$ is a Baire space. Following a definition of A.R. Todd from [a3], the topologies $\rho$ and $\tau$ are $S$-related if for any subset $A$ of $X$, $\operatorname { Int } _ { \rho } A$ (the interior of $A$ with respect to the topology $\rho$) is non-empty if and only if $\operatorname { Int } _ { \tau } A$ is non-empty. If the topologies $\tau$ and $\rho$ are $S$-related, then $\tau$ has the Slobodnik property if and only if $( X , \rho )$ is a Baire space, and this is the case if and only if $( X , \tau )$ is a Baire space.

Let $f$ be a function on $X$ of the first Baire class in the topology $\tau$, i.e. $f$ is a pointwise limit of a sequence of $\tau$-continuous functions (cf. also Baire classes). A very general problem emerges: How large can the set of all $\rho$-continuity points of $f$ be? If $\tau$ has the Slobodnik property, then $f$ is $\rho$-continuous at all points of $X$ except at a set of $\rho$-first category. This theorem generalizes Slobodnik's theorem from [a2]: Any limit of a sequence of separately continuous functions on the Euclidean space $\mathbf{R} ^ { 2 }$ is continuous on $\mathbf{R} ^ { 2 }$, except at a set of the first category. Notice that separately continuous functions are of the first Baire class on $\mathbf{R} ^ { 2 }$, that a function $f$ is separately continuous on $\mathbf{R} ^ { 2 }$ exactly when it is continuous in the finer crosswise topology on $\mathbf{R} ^ { 2 }$ (a set $G \subset \mathbf{R} ^ { 2 }$ is open in this topology if for any $z = ( z _ { 1 } , z _ { 2 } ) \in G$ there is a $\delta > 0$ such that the "cross"

\begin{equation*} K ( z , \delta ) : = \left\{ \begin{array}{l} {} & { t _ { i } = z _ { i }, }\\{ ( t _ { 1 } , t _ { 2 } ) :} &{ | z _ { j } - t _ { j } | < \delta, }\\{} & { i , j = 1,2 , i \neq j }\end{array} \right\} \end{equation*}

is a subset of $G$), 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