## Operations on convergences.(English)Zbl 0938.54005

A (sequential) convergence in a set $$X$$ is a map $$G:X^N\rightarrow 2^X$$. If $$\xi \in G(\{ \xi _n\})$$, then the sequence $$\{ \xi _m\}$$ $$G$$ converges to $$\xi \in X$$. The usual basic axioms of convergence are: $$H$$ (uniqueness of limits), $$S$$ (constants), $$F$$ (subsequences), $$U$$ (Urysohn). If $$X$$ is a group, then the following additional axioms are considered: $$L$$ (compatibility), $$K$$ and $$N$$ (in terms of summable subsequences of a zero sequence), $$D$$ and $$Y$$ (in terms of quasi diagonals of a sequence of zero sequences). If $$X$$ is a linear space, then $$N'$$ and $$M$$ (in terms of products of a zero sequence and a sequence of scalars) play an important role in applications to functional analysis and measure theory. For a family $$\{ G_a$$; $$a\in A\}$$ of convergences, four operations are defined in a natural way: intersection, product, union, quotient. The authors study various cases when the operation in question preserves or does not preserve a given axiom of convergence. Interesting results, examples, and counterexamples (related to function spaces) are presented.
Sample result: $$D$$ and $$Y$$ are preserved under countable intersections.

### MSC:

 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 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.) 54B99 Basic constructions in general topology 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)