arXiv subjects
Publications and source records attributed to Robson A. Figueiredo.
We prove a definable version of the Whitney embedding theorem for abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p<\infty$, namely: every abstract-definable $\mathcal{C}^p$ manifold is abstract-definable $C^p$ embedded into $R^N$, for some positive integer $N$. As a consequence, we show that every abstract-definable $\mathcal{C}^p$ manifold has a compatible $\mathcal{C}^{p+1}$ atlas.
In this work we deal with the preservation by $G_δ$-refinements. We prove that for $\mathrm{SP}$-scattered spaces the metacompactness, paralindelöfness, metalindelöfness and linear lindelöfness are preserved by $G_δ$-refinements. In this context we also consider some other generalizations of discrete spaces like $ω$-scattered and $N$-scattered. In the final part of this paper we look at a question of Juhász, Soukup, Szentmiklóssy and Weiss concerning the tighness of the $G_δ$-refinement of a $σ$-product.