The Shape of Generating Families

Abstract

The topology of a space $X$ is generated by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq X$ is closed in $X$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. A space $X$ is a $k$-space (respectively, sequential) if its topology is generated by the collection of all compact subsets (respectively, convergent sequences) of $X$. Relations are defined to capture the notion of a space being a $k$-space or sequential. The structure (or `shape') under the Tukey order of these relations applied to separable metrizable spaces is examined. For the $k$-space case the initial structure is completely determined, and the cofinal structure is shown to be highly complex. In the sequential case, however, the entire shape is determined. It follows that the number of Tukey types in the sequential case lies between $\aleph_0$ and $\mathfrak{c}$, is equal to $\aleph_0$ precisely when $\mathfrak{c} < \aleph_{ω_1}$, and is equal to $\mathfrak{c}$ if and only if $\mathfrak{c}$ is a fixed point of the aleph function, necessarily of uncountable cofinality.