## Applications of Set Theory to Analysis and Topology |

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ALMOST DISJOINT SETS Alster axiom of choice Banach spaces Borel measure Borel set Cardinal functions chapter choose clopen closed map closed sets closed subset closure cofinal collection collectionwise Hausdorff collectionwise normal compact Hausdorff space compact set COMPACT SUBSETS complete regularity Computer Science construction contained continuous function continuum hypothesis Corollary countable chain condition countable set countable space countable subset countably compact countably infinite countably metacompact countably paracompact Moore cr-finite define denote dense subset discrete discrete measure discrete topology disjoint open sets DOCTOR OF PHILOSOPHY Dowker space Eberlein compact every countable example exists extremally disconnected finite measure Fleissner functionally Katetov Hausdorff space hereditarily separable homeomorphic However implies induction Kunen Lebesgue measure lemma Let X limit ordinal limit point locally compact locally compact space locally finite locally finite cover locally finite open Lusin measure Lusin space Lusin1 s theorem M. L. Wage Martin's axiom Mary Ellen Rudin Math measurable function measure on X measure zero metric space metrizable Michael Lee n e co neco neighborhood neighborhood base non-metrizable non-normal Moore space non-normal space normal Moore space normal space open cover open subset order topology order type ordinal in co outer regular pairwise paracompact Moore space paracompact space partial order perfect spaces product topology Proof of Theorem properties R. H. Bing real line regular regular space Riesz measure separated by disjoint set theoretic sets in X sets of finite Since Souslin space X subset of X subspace Suppose Symbolic Logic thesis topological spaces topology total ordering ultrafilter uncountable uncountable set University of Wisconsin weakly WEAKLY COMPACT X co X is countably