**"Compact Generators"**
#### Gartside, PaulA subset $G$ of the set $C(X)$ of all continuous, real-valued functions on a Tychonoff space $X$ is a \emph{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.
Joint work with Jeremiah Morgan and Alex Yuschik. |