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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]]).
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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.
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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"
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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 $X$ is Baire if and only if the intersection of each countable family of dense open sets in $X$ is dense (cf. also [[Dense set|Dense set]]).
  
<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>
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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 [[#References|[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.
  
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.
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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|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 [[#References|[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 $\delta &gt; 0$ such that the  "cross"  
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\begin{equation*} K ( z , \delta ) : = \left\{ \begin{array}{l} {} &amp; { t _ { i } = z _ { i }, }\\{ ( t _ { 1 } , t _ { 2 } ) :} &amp;{ | z _ { j } - t _ { j } | &lt; \delta, }\\{} &amp; { i , j = 1,2 , i \neq j }\end{array} \right\} \end{equation*}
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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 [[#References|[a1]]].
 
A more detailed investigation of the Slobodnik property and related notions can be found in [[#References|[a1]]].
  
 
====References====
 
====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>
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<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>

Revision as of 16:59, 1 July 2020

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
How to Cite This Entry:
Slobodnik property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Slobodnik_property&oldid=50313
This article was adapted from an original article by J. Lukeš (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article