In this dissertation, some properties of evaluation maps on groups of self-homeomorphisms will be studied. In particular, the following question is considered: For what type of spaces, X, and what topologies, T $\sp*$, on a group, G, of self-homeomorphisms on X, will give us that, for each x $\in$ X, the evaluation map, E$\sb{\rm x}$: (G, T $\sp*$) $\to$ X, defined by E$\sb{\rm x}$(g) = g(x), is an open map?

In 1965, E. G. Effros published a very important result which gave one answer to the preceding question. One form of Effros' Theorem is: If X is a compact, homogeneous, metric space, then for all x $\in$ X, the evaluation map, E$\sb{\rm x}$: (H (X), T $\sb{\bf co}$) $\to$ X, defined by E$\sb{\rm x}$(g) = g(x), is an open map. Here H (X) is the collection of all self-homeomorphisms on X and T $\sb{\bf co}$ is the compact-open topology.

The main results of this dissertation are as follows; first, the introduction of even homogeneity as a generalization of the concept of ULH spaces and the development of some of its properties; and second, the use of even homogeneity and Effros' Theorem to extend Effros's Theorem to the case where X is an uncountable product of compact, homogeneous, metric spaces.