| MSC: | 54C35 | Function spaces [See also 46Exx, 58D15] |
| 54C05 | Continuous maps | |
| 54D30 | Compactness |
Abstract:
Let $(X,\tau)$, $(Y,\sigma)$ be topological spaces, and let be $\emptyset\neq\ga\subseteq\Pmw{X}$.
We consider the set-open topology $\taga$ for $Y^X$ or for $C(X,Y)$, generated by the family $\ga$,
and we assume that $\tau_p\subseteq \taga$ holds, where $\tau_p$ denotes the pointwise topology.
For $H\subseteq C(X,Y)$ we want to characterize the $\taga$-compactness of $H$.
We will need the condition that $H$ is evenly continuous on each $A\in\ga$.
Hence we consider both sets $C(X,Y)$ and $C(A,Y)$ and of course we can link these spaces
by the map $q_A:q_A(f):=f_{|A}$, the restriction of $f$ to the subspace $(A,\tau_{|A})$ of $(X,\tau)$.
So we have $q_A:C(X,Y)\to C(A,Y)$.
Using these maps, we give a new and interesting proof of a ''final'' kind of the Ascoli theorem,
as former derived by use of hyperspaces in \cite{ren-rb_nz2004}.
Most notions used here are standard and explanations can be found in standard books on general topology
such as \cite{ren-dugu}, \cite{ren-engelking}, \cite{ren-kelley}. Concerning some more special notions
we refer to \cite{ren-bapode}, more explanations can be found in \cite{ren-pop2}
and \cite{ren-rb_nz2004}, too.