Local convexity of twisted sums. (English) Zbl 0538.46006

Domaǹski, Pawel

Rend. Circ. Mat. Palermo, II. Ser. (to appear).

This article is a completion of the paper reviewed in Zbl 0538.46005 above. TSC-spaces (definitions see above) are characterized in different ways, for example a lcs Z is a TSC-space iff \(Z\in S(\ell_{\infty}(A))\) for every set A. An arbitrary product or countable direct sum of metrizable lcs is a TSC-space iff it belongs to S(K), dim K\(=1.\)
New permanence properties for the classes S(Y) and TSC-spaces are proved and all these properties are summarized. Existence of (homogeneous) continuous (or continuous at zero) sections for every quotient map q, \(\ker q\simeq Y, im q\simeq Z\) is shown for some tvs Y, Z. This allows to improve the method of constructing all twisted sums of tvs Y, Z given in the previous paper.

MSC:

46A19	Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
46A04	Locally convex Fréchet spaces and (DF)-spaces

Keywords:

three space problem; complemented; continuous section; continuous selection; TSC-spaces; permanence properties; twisted sums

Citations:

Zbl 0538.46005