On extensions of convergences (Q2774509)

Let \(X\) be a linear space. A sequential convergence in \(X\) is a subset of \(X^N\times X\) defining which sequence of points converges to which point. The usual FLUSH axioms (subsequences, sequential continuity of the linear operations, the Urysohn axiom, constants, the uniqueness of limits) are considered. Convergences in \(X\) are partially ordered by inclusion. NEWLINENEWLINENEWLINELet \(\mathcal G \subset X^N\times X\) be a convergence in \(X\) satisfying some of the axioms (usually all five). The author introduced natural notions like outer sequence, essentially outer sequence, strongly outer sequence (for \(\mathcal G\)) in order to measure ``how much freedom'' we have in enlarging \(\mathcal G\) to a convergence \(\mathcal G'\) so that a given sequence \((x_n)\) does not converge to a point \(x\) under \(\mathcal G\) and \((x_n)\) does converge to \(x\) under \(\mathcal G'\) and \(\mathcal G'\) satisfies the original axioms satisfied by \(\mathcal G\). Besides interesting results, the paper contains a number of illustrating examples.