An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic `1 Banach space. The construction of this operator relies on the existence of transfinite c0-spreading models in the dual of the space.

We prove that hereditarily disconnected countably compact groups are zero-dimensional. This gives a strongly positive answer to a question of Shakhmatov. We show that hereditary or total disconnectedness yields zerodimensionality in various classes of pseudocompact groups.

Several classes of hereditarily normal spaces are characterized in terms of extending upper semi-continuous compact-valued mappings. The case of controlled extensions is considered as well. Applications are obtained for real-valued semi-continuous functions.

computing devices (the characterization was then improved and more exten-sively explained by W. Sieg in [34], [35]) and by N. De Pisapia in [11] to model abstractneural nets. The cortex models presented in the papers quoted above are described in the language ofhereditarily finite sets in Sections 2 and 3 of the present paper, where the links betweenthe mentioned cortex aspects ...

A Hereditarily Indecomposable asymptotic ℓ 2 Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W.T. Gowers.

Abstract A subgroup of a group G is said to be hereditarily -permutable with B , if $AB^x = B^xA$ for some element $x \in \langle A, \rangle $ . in every In this paper, we investigate the structure finite all its Schmidt subgroups -permutable.

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