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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)
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