By W. W. Comfort

A series is a estate, commonly regarding concerns of cardinality, of the kinfolk of open subsets of a topological house. (Sample questions: (a) How huge a fmily of pairwise disjoint open units does the distance admit? (b) From an uncountable kinfolk of open units, can one continually extract an uncountable subfamily with the finite intersection estate. This monograph, that's in part clean examine and in part expository (in the feel that the authors co-ordinate and unify disparate effects bought in different diversified international locations over a interval of a number of many years) is dedicated to the systematic use of infinitary combinatorial tools in topology to procure effects bearing on chain stipulations. The combinatorial instruments constructed through P. Erdös and the Hungarian tuition, by means of Erdös and Rado within the Sixties and by way of the Soviet mathematician Shanin within the Forties, are enough to address many typical questions bearing on chain stipulations in product areas.

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**Example text**

Spaces. ' Hewitt (1948) defined pseudocompact spaces (as those completely regular Hausdorff spaces on which every continuous real-valued function is bounded) and gave several characterizations based on the ring C(X) and the inclusion X a px. That Hewitt's spaces are exactly those for which every locally finite family of open subsets is finite was shown by Glicksberg (1952); see also Mardesifc & Papifc (1955) and Bagley, Connell & McKnight (1958). Isbell (1964) (page 135) introduced pseudo-a-compact spaces for a > co as those (uniform) spaces whose covering character in the fine uniformity is at most a; it is noted by Noble (1969a) (page 389) that Isbell's pseudo-a-compact spaces are exactly ours.

A} is as required. Let Aa = Ac\[a,,aff+ Jfor o < cf(a), and set B = {a < cf(a) :Aa ^ 0 } . If | J5| < A, then since A = 2 or is regular there is a < cf(a) such that | A. \ = X and we have and if | B \ = X then (X < cf(a) and Ua is defined for a < cf(a) and) we have n K c n Ua = 0 . Reduction to compact Hausdorff spaces For use in describing the calibre properties of compact spaces we introduce the relations < and = on the class of spaces and we show that every space X is = -equivalent to its (compact) Gleason space G(X).

Spaces. ' Hewitt (1948) defined pseudocompact spaces (as those completely regular Hausdorff spaces on which every continuous real-valued function is bounded) and gave several characterizations based on the ring C(X) and the inclusion X a px. That Hewitt's spaces are exactly those for which every locally finite family of open subsets is finite was shown by Glicksberg (1952); see also Mardesifc & Papifc (1955) and Bagley, Connell & McKnight (1958). Isbell (1964) (page 135) introduced pseudo-a-compact spaces for a > co as those (uniform) spaces whose covering character in the fine uniformity is at most a; it is noted by Noble (1969a) (page 389) that Isbell's pseudo-a-compact spaces are exactly ours.