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Register a new user(Metrically) quarter-stratifiable spaces and their applications in the theory of separately continuous functions
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Taras Banakh
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We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions.
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Let $fcolonmathbb{R}^2tomathbb{R}$. The notions of feebly continuity and very feebly continuity of $f$ at a point $langle x,yrangleinmathbb{R}^2$ were considered by I. Leader in 2009. We study properties of the sets $FC(f)$ (respectively, $VFC(f)supset FC(f)$) of points at which $f$ is feebly continuous (very feebly continuous). We prove that $VFC(f)$ is densely nonmeager, and, if $f$ has the Baire property (is measurable), then $FC(f)$ is residual (has full outer Lebesgue measure). We describe several examples of functions $f$ for which $FC(f) eq VFC(f)$. Then we consider the notion of two-feebly continuity which is strictly weaker than very feebly continuity. We prove that the set of points where (an arbitrary) $f$ is two-feebly continuous forms a residual set of full outer measure. Finally, we study the existence of large algebraic structures inside or outside various sets of feebly continuous functions.
In this paper, we characterize stratifiable (or semi-stratifiable) spaces, and monotonically countably paracompact (or monotonically countably metacompact) spaces by expansions of locally upper bounded semi-continuous poset-valued maps. These extend earlier results for real-valued Locally bounded functions.
Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.
The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $mathbb R^m$ this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function $f:mathbb R^mtomathbb R$ is of the $(m-1)$-th Baire class. In this paper we prove that every linearly continuous function $f:mathbb R^mtomathbb R$ is of the first Baire class. Moreover, we obtain the following result. If $X$ is a Baire cosmic topological vector space, $Y$ is a Tychonoff topological space and $f:Xto Y$ is a Borel-measurable (even BP-measurable) linearly continuous function, then $f$ is $F_sigma$-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on $mathbb R^m$. In the final part of the article we prove that any $F_sigma$-measurable function $f:partial Uto mathbb R$ defined on the boundary of a strictly convex open set $Usubsetmathbb R^m$ can be extended to a linearly continuous function $bar f:Xto mathbb R$. This fact shows that in the ``descriptive sense the linear continuity is not better than the $F_sigma$-measurability.
A function $f:Xto Y$ between topological spaces is called $sigma$-$continuous$ (resp. $barsigma$-$continuous$) if there exists a (closed) cover ${X_n}_{ninomega}$ of $X$ such that for every $ninomega$ the restriction $f{restriction}X_n$ is continuous. By $mathfrak c_sigma$ (resp. $mathfrak c_{barsigma}$) we denote the largest cardinal $kappalemathfrak c$ such that every function $f:Xtomathbb R$ defined on a subset $Xsubsetmathbb R$ of cardinality $|X|<kappa$ is $sigma$-continuous (resp. $barsigma$-continuous). It is clear that $omega_1lemathfrak c_{barsigma}lemathfrak c_sigmalemathfrak c$. We prove that $mathfrak plemathfrak q_0=mathfrak c_{barsigma}=min{mathfrak c_sigma,mathfrak b,mathfrak q}lemathfrak c_sigmalemin{mathrm{non}(mathcal M),mathrm{non}(mathcal N)}$. The equality $mathfrak c_{barsigma}=mathfrak q_0$ resolves a problem from the initial version of the paper.
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