CONVERGENCE OF EVENLY CONTINUOUS NETS IN GENERAL FUNCTION SPACES

Abstract

For topological spaces $X$, $Y$ let $Y^X$, $C(X,Y)$ denote the sets of all functions from $X$ to $Y$ and of all continuous functions from $X$ to $Y$ respectively. We consider nets of functions from these spaces which are evenly continuous or pointwise equicontinuous, and for such nets the relationship between pointwise convergence and topological convergence is studied. We find that in $Y^X$ for an evenly continuous net pointwise convergence (to a function from $Y^X$) implies topological convergence. Conversely, if $Y$ is an uniform space and $(f_i)$ a pointwise equicontinuous net then $\mathrm{\Gamma }{f}_1$ lim inf $\mathrm{\Gamma }{f}_i$ implies the pointwise convergence of $(f_i)$ to $f$. By an example is shown that in the second assertion the equicontinuity of $(f_i)$ cannot be replaced by even continuity. As corollaries of our results we get some results of T. Neubrunn and Ľ. Holà [6] and of G. Beer [1] respectively. Moreover, a relationship to the lower semi-finite graph-topology is established.