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Register a new userReturning functions with closed graph are continuous
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A function $f:Xto mathbb R$ defined on a topological space $X$ is called returning if for any point $xin X$ there exists a positive real number $M_x$ such that for every path-connected subset $C_xsubset X$ containing the point $x$ and any $yin C_xsetminus{x}$ there exists a point $zin C_xsetminus{x,y}$ such that $|f(z)|le max{M_x,|f(y)|}$. A topological space $X$ is called path-inductive if a subset $Usubset X$ is open if and only if for any path $gamma:[0,1]to X$ the preimage $gamma^{-1}(U)$ is open in $[0,1]$. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible space. We prove that a function $f:Xto mathbb R$ defined on a path-inductive space $X$ is continuous if and only of it is returning and has closed graph. This implies that a (weakly) Swic atkowski function $f:mathbb Rtomathbb R$ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscibed to Lviv Scottish Book.
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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.
Let $Sigma (X,mathbb{C})$ denote the collection of all the rings between $C^*(X,mathbb{C})$ and $C(X,mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/$z$-ideals/$z^circ$-ideals in the rings $P(X,mathbb{C})$ in $Sigma(X,mathbb{C})$ and in their real-valued counterparts $P(X,mathbb{C})cap C(X)$. It is shown that the structure space of any such $P(X,mathbb{C})$ is $beta X$. We show that for any maximal ideal $M$ in $C(X,mathbb{C}), C(X,mathbb{C})/M$ is an algebraically closed field. We give a necessary and sufficient condition for the ideal $C_{mathcal{P}}(X,mathbb{C})$ of $C(X,mathbb{C})$ to be a prime ideal, and we examine a few special cases thereafter.
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.
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.
A proper ideal $I$ in a commutative ring with unity is called a $z^circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a completely $F$-regular topological space $X$, let $C(X,F)$ be the ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ the aggregate of all those functions which are bounded over $X$. An explicit formula for all the $z^circ$-ideals in $A(X,F)$ in terms of ideals of closed sets in $X$ is given. It turns out that an intermediate ring $A(X,F) eq C(X,F)$ is never regular in the sense of Von-Neumann. This property further characterizes $C(X,F)$ amongst the intermediate rings within the class of $P_F$-spaces $X$. It is also realized that $X$ is an almost $P_F$-space if and only if each maximal ideal in $C(X,F)$ is $z^circ$-ideal. Incidentally this property also characterizes $C(X,F)$ amongst the intermediate rings within the family of almost $P_F$-spaces.
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