We explore almost-normality in Isbell-Mr\'owka spaces and some related concepts. use forcing to provide an example of almost-normal not normal almost disjoint family, the concept semi-normality spaces, define strongly $(\aleph_0, <\mathfrak c)$-separated families prove generic existence completely separable c)$-almost assuming $\mathfrak s=\mathfrak c$ b=\mathfrak c$. also a Tychonoff pseudocom...

We obtain a characterization of left perfect rings via superstability the class flat modules with pure embeddings. $\mathbf{Theorem.}$ For ring $R$ following are equivalent. - is perfect. The $R$-modules embeddings superstable. There exists $\lambda \geq (|R| + \aleph_0)^+$ such that has uniqueness limit models cardinality $\lambda$. Every model in $\Sigma$-cotorsion. A key step our argum...

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, Tsirelson space Figiel and Johnson \cite{FJ}. Second, using results first part, give general Banach spaces. Third, study model-theoretic dividing lines in spaces their theories. In~particular, show: (1) $M_{\mathcal{T}}$ \emph{non independence property} (NIP); (2) every that is $\ale...

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\not= 2<\infty$. This solves a problem A. Pietsch's 1978 book Operator Ideals. The proof is quite from other methods of producing space bounded operators on space; particular, not contained strictly singular and yet do contain projections onto subspaces non Hilbertian. give criterion for w...

this paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

We construct a homogeneous connected Polish space X on which no א0-bounded topological group acts transitively. In fact, X is homeomorphic to a convex subset of Hilbert space `.