View source for Slobodnik property

Recall that a [[Baire space|Baire space]] is a [[Topological space|topological space]] in which every non-empty open subset is of the second category in itself (cf. also [[Category of a set|Category of a set]]). A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303801.png" /> is Baire if and only if the intersection of each countable family of dense open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303802.png" /> is dense (cf. also [[Dense set|Dense set]]).

In what follows, consider a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303803.png" /> equipped with two topologies, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303805.png" />, and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303806.png" /> is finer than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303807.png" />. A topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303808.png" /> has the Slobodnik property if the intersection of each countable family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s1303809.png" />-open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038010.png" />-dense sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038011.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038012.png" />-dense. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038013.png" /> has the Slobodnik property, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038014.png" /> is a Baire space. Following a definition of A.R. Todd from [[#References|[a3]]], the topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038016.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038018.png" />-related if for any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038021.png" /> (the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038022.png" /> with respect to the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038023.png" />) is non-empty if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038024.png" /> is non-empty. If the topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038026.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038027.png" />-related, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038028.png" /> has the Slobodnik property if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038029.png" /> is a Baire space, and this is the case if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038030.png" /> is a Baire space.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038031.png" /> be a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038032.png" /> of the first Baire class in the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038033.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038034.png" /> is a pointwise limit of a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038035.png" />-continuous functions (cf. also [[Baire classes|Baire classes]]). A very general problem emerges: How large can the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038036.png" />-continuity points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038037.png" /> be? If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038038.png" /> has the Slobodnik property, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038039.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038040.png" />-continuous at all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038041.png" /> except at a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038042.png" />-first category. This theorem generalizes Slobodnik's theorem from [[#References|[a2]]]: Any limit of a sequence of separately continuous functions on the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038043.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038044.png" />, except at a set of the first category. Notice that separately continuous functions are of the first Baire class on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038045.png" />, that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038046.png" /> is separately continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038047.png" /> exactly when it is continuous in the finer crosswise topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038048.png" /> (a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038049.png" /> is open in this topology if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038050.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038051.png" /> such that the  "cross" 

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038052.png" /></td> </tr></table>

is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130380/s13038053.png" />), and that the crosswise topology has the Slobodnik property.

A more detailed investigation of the Slobodnik property and related notions can be found in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Lukeš,   J. Malý,   L. Zajíček,   "Fine topology methods in real analysis and potential theory" , ''Lecture Notes in Mathematics'' , '''1189''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.G. Slobodnik,   "Expanding system of linearly closed sets"  ''Mat. Zametki'' , '''19'''  (1976)  pp. 61–84  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Todd,   "Quasiregular, pseudocomplete, and Baire spaces"  ''Pacific J. Math.'' , '''95'''  (1981)  pp. 233–250</TD></TR></table>