By studying dimensional types of metric scattered spaces, we consider the wider class of metric σ-discrete spaces. Applying techniques relevant to this wider class, we present new proofs of some embeddable properties of countable metric spaces in such a way that they can be generalized onto uncountable metric scattered spaces. Related topics are also explored, which gives a few new results.

We show that there are stationary subsets of uncountable spaces which do not reflect.

A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal $a$ has a ternary partition (see Section 1, Definition 2) then the Russell cardinal $a+2$ fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell...

In this short note we show that for fuzzy sets of type II the additive rule for cardinalities holds true. The proof of this result requires an application of approximate reasoning as means of inference by use of the entailment principle.

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{\xi }$ generated by subsequences $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$, respectively, of the natural Schauder basis $\left(e_n^{\xi }\right)$ of $X^{\xi }$ are isomorphic if and only if $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$ are equivalent. Further, $X^{\xi }$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $\left(e_n^{\xi }\right)$. It is also shown that there exists a complemented subspace spanned by a block basis of $\left(e_n^{\xi }\right)$, which is not isomorphic to a subspace generated by a subsequence of $\left(e_n^{\zeta }\right)$, for every $0\le \zeta \le \xi$....

The following question is analyzed under the assumption that the Axiom of Choice fails badly: Given a countable number of pairs of socks, then how many socks are there? Surprisingly this number is not uniquely determined by the above information, thus giving rise to the concept of Russell-cardinals. It will be shown that: • some Russell-cardinals are even, but others fail to be so; • no Russell-cardinal is odd; • no Russell-cardinal is comparable with any cardinal of the form ${\aleph }_{\alpha }$ or $2^{{\aleph }_{\alpha }}$; • finite sums...

It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself such that if Y is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$, then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself there is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$ on which the operator is an isomorphism.