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Register a new userBaire category properties of function spaces with the Fell hypograph topology
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For a Tychonoff space $X$ and a subspace $Ysubsetmathbb R$, we study Baire category properties of the space $C_{downarrow F}(X,Y)$ of continuous functions from $X$ to $Y$, endowed with the Fell hypograph topology. We characterize pairs $X,Y$ for which the function space $C_{downarrow F}(X,Y)$ is $infty$-meager, meager, Baire, Choquet, strong Choquet, (almost) complete-metrizable or (almost) Polish.
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We prove that for a stratifiable scattered space $X$ of finite scattered height, the function space $C_k(X)$ endowed with the compact-open topology is Baire if and only if $X$ has the Moving Off Property of Gruenhage and Ma. As a byproduct of the proof we establish many interesting Baire category properties of the function spaces $C_k(X,Y)={fin C_k(X,Y):f(X)subset{*_Y}}$, where $X$ is a topological space, $X$ is the set of non-isolated points of $X$, and $Y$ is a topological space with a distinguished point $*_Y$.
In this paper, we intend to show that under not too restrictive conditions, results much stronger than the one obtained earlier by Hejduk could be established in category bases.
W. Hurewicz proved that analytic Menger sets of reals are $sigma$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgoz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with $omega$-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.
Let $mathcal{I}$ be a meager ideal on $mathbf{N}$. We show that if $x$ is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of $x$ which preserve the set of $mathcal{I}$-cluster points of $x$ is topologically large if and only if every ordinary limit point of $x$ is also an $mathcal{I}$-cluster point of $x$. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. textbf{263} (2019), 221--229]. As an application, if $x$ is a sequence with values in a first countable compact space which is $mathcal{I}$-convergent to $ell$, then the set of subsequences [resp. permutations] which are $mathcal{I}$-convergent to $ell$ is topologically large if and only if $x$ is convergent to $ell$ in the ordinary sense. Analogous results hold for $mathcal{I}$-limit points, provided $mathcal{I}$ is an analytic P-ideal.
We use lexicographic products to give examples of compact spaces of first Baire class functions on a compact metric space that cannot be represented as spaces of functions with countably many discontinuities.
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