Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Baire Category of Subsequences and Permutations which preserve Limit Points
223
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
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 $*_Y$.
The Omitting Types Theorem in model theory and the Baire Category Theorem in topology are known to be closely linked. We examine the precise relation between these two theorems. Working with a general notion of logic we show that the classical Omitting Types Theorem holds for a logic if a certain associated topological space has all closed subspaces Baire. We also consider stronger Baire category conditions, and hence stronger Omitting Types Theorems, including a game version. We use examples of spaces previously studied in set-theoretic topology to produce abstract logics showing that the game Omitting Types statement is consistently not equivalent to the classical one.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
