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Sets and Functions
Infinite Sets and Transfinite Numbers
Some Familiar Topological Spaces and Basic Topological
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7Vspace absolute value metric accumulation point basis bounded subset called cardinality Cauchy sequence Chapter closed and bounded closed sets closed subset closure cluster point collection of non-empty compact space compact subset compactification complete metric space completely regular space contains continuous function countably compact definition denoted discrete space discrete topology disjoint open sets element equivalent filter finite ordinal numbers following theorem Give an example given Hausdorff space Hint for proof homeomorphic intersection Lemma locally connected neighborhood non-empty collection non-empty topological spaces open cover open interval open sets open subset order topology positive integers Problem product space product topology projection functions prove pseudocompact r-ball rational numbers real line Recall relative topology relative topology inherited sequentially compact set is open Show space is compact space need spaces and let Sr(x sub-basis subspace topological space totally bounded unit interval usual topology Xi and X2 Xi X X2