D-3 - Fréchet and Sequential Spaces

Abstract

It is possible that the topology of a space X is fully determined by the collection of all convergent sequences in the space. Such spaces are called Fréchet spaces (synonymously, Fréchet–Urysohn spaces) or sequential, depending on how the topological closure can be obtained from convergent sequences. Thus, the space X is Fréchet, if the closure of any set agrees with its sequential closure, and sequential, if the closure of any set is obtained by the iteration of a sequential closure. Every first-countable space is Fréchet and every Fréchet space is sequential. The reverse implications do not hold. The class of sequential spaces is closed undertaking quotients and disjoint topological sums, and consequently under inductive limits. Closed and open subspaces of sequential spaces are sequential too, but in general, sequentiality is not hereditary. A Hausdorff space is Fréchet if and only if (iff) it is an image of a topological sum of convergent sequences under a pseudo-open quotient map. In contrast to sequential spaces, the class of Fréchet spaces is hereditary, and a sequential space is Fréchet if it is hereditary sequential. Compactness or some of its weaker forms has a strong influence on sequential spaces and vice versa.