Countable products of generalized countably compact spaces

In this paper two ways of generalizing compactness are studied. We may consider various types of refinements of open covers, such as countable open refinements, locally finite open refinements, etc. In another direction, countably compact spaces may be characterized as having the property that any sequence has a cluster point. Spaces which require that certain sequences have cluster points, such as ^-spaces, mJ-spaces, and g-spaces, will be referred to as generalized countably compact spaces.