Let ?X indicate the Freudenthal compactification of a rimcompact, completely regular Hausdorff spaceX. In this paper the spacesY which satisfyX?Y??X are characterized. From this a characterization of whenX lies between its locally compact partL(X) and ?(L(X)) follows. Such spaces necessarily possess a compactification aX for whichClaX(aX-X) is 0-dimensional. Conditions, including those internal toX, are provided which are necessary and sufficient for this property to hold.