The coreflective hull of \(\mathcal{U}\)-Fréchet spaces (Q2024077)

The present paper adopts a generalized concept of sequential convergence by defining an element \(x\) of a topological space to be an \(\mathcal{U}\)-limit of \(\{x_n\}_{n \in \mathbb{N}}\) if for each open neighborhood \(V\) of \(x\) we have \(\{n \mid x_n \in V\} \in \mathcal{U}\) for an arbitrary free filter \(\mathcal{U}\) on \(\mathbb{N}\) instead of the usual Fréchet filter. \(\mathcal{U}\)-sequential spaces in that sense are shown to be closed under quotients and sums and hence form a bi-coreflective subcategory of \textit{Top}. Moreover, for a free ultrafilter \(\mathcal{U}\) on \(\mathbb{N}\), the assignment of \(\mathcal{U}\)-limits gives rise to a lower limit operator \(l\) [\textit{H. Herrlich}, Trans. Am. Math. Soc. 146, 203--210 (1969; Zbl 0194.54201)], which captures the \(\mathcal{U}\)-sequential spaces as the \(l\)-separated spaces and thereby reproves bi-coreflectivity. An example shows that this limit operator is not idempotent, which leads to the consideration of \(\mathcal{U}\)-Fréchet spaces, whose coreflective hull is the category of \(\mathcal{U}\)-sequential spaces.