René Bartsch
Harry Poppe

### Compactness in function spaces with splitting topologies

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 66, 69 - 73 (2011)

 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.

Keywords:   Function spaces, Continuous maps, Compactness

karin.martin@uni-rostock.de
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