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Abstract. Let m > n0 and X = ]l¡elX¡. Then X is [n0, m]-compact if and only if ¡¡¡eJX¡ is [K0, m]-compact for all J c / with |y| < 22™. Let m > Kq, 0{: £ < m) a net in X,p e X, and <$ e /3(m). Then/? = «¡) -lim£

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Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then A-is a continuous perfect image of a subspace of ßD which contains D if and only if A" is regular. 1. Introduction. In [B,], A. Bernstein makes the following definitions. Let ^D be an ultrafilter over the positive integers N, let A' be a topological space, (xn:n G N) a sequence of points of X, and x G X. Then x is a ty-limit point of (xn: n G A/) provided that if t/ is open in X with x E U, then {«: *" G Í/} G 6D ; in this case we write x = ^-lim"^0O jc". If every sequence in X has a ty-limit point, then X is ^-compact. In [GS], John Ginsburg and the present author showed that a Hausdorff space X has all of its powers countably compact if, and only if, it iŝ D-compact, for some ^ in ßN \ N. In this paper, we generalize the notion of ^-limit to nets indexed by an arbitrary infinite cardinal. We show that any topological space is characterized by its generalized ^D-limits, and that a number of topological properties can be described by the use of ultrafilter invariants.

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