Compact generators
Volume 259 / 2022
Abstract
A subset $G$ of the set $C(X)$ of all continuous, real-valued functions on a Tikhonov space $X$ is a generator if, whenever $x$ is a point of $X$ not in a closed set $C$, there is a $g$ in $G$ such that $g(x) \notin \overline {g(C)}$.
Considering $C(X)$ with the compact-open topology, and every generator as a subspace: $X$ is metrizable if and only if it is a $k$-space and has a compact generator.
Considering $C(X)$ with the topology of pointwise convergence: $X$ has a compact generator if and only if $X$ is Eberlein–Grothendieck; $X$ has a generator homeomorphic to a supersequence (one-point compactification of a discrete space) if and only if $X$ has a $\sigma $-point-finite base of cozero sets; $X$ has a generator homeomorphic to the continuous image of the one-point compactification of the disjoint sum of finite powers of a supersequence if and only if $X$ has a $\sigma $-point-finite almost subbase; if $X$ has a Lindelöf generator, then it is countably tight, but $X^2$ need not be countably tight; and a space may have all finite powers countably tight but have no Lindelöf generator.
