## Local convexity of twisted sums.(English)Zbl 0538.46006

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

Zbl 0538.46005