Download PDF by John Stuart Mill: A System of Logic Ratiocinative and Inductive, Part II (The

By John Stuart Mill

ISBN-10: 0710075065

ISBN-13: 9780710075062

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2) Not every completely regular space is Tychonoff–compact. 44 Show that in the realm of pseudometric spaces as well as in the realm of Hausdorff spaces the following conditions are equivalent: (1) Compact = Alexandroff–Urysohn–compact. (2) Ultrafilter–compact = Alexandroff–Urysohn–compact. (3) AC. E 8. Show the equivalence of: (1) Every space with a finite topology is Alexandroff–Urysohn–compact. (2) AC. E 7. E 9. 45 Call a topological space X • Lindel¨ of, if every open cover of X contains an at most countable subcover of X.

Weierstrass–compact = compact. 29. 35 In the realm of pseudometric spaces the following conditions are equivalent: 1. Fin. 2. Weierstrass–compact = sequentially compact. 3 Concepts Split Up: Compactness 39 Proof. (1) ⇒ (2) Every Weierstrass–compact space is sequentially compact. For the converse, consider an infinite subset A of a sequentially compact space X. By (1), there exists an injective sequence (an ) in A. By sequential compactness of X, some subsequence of (an ) converges to some x ∈ X.

E 9. 45 Call a topological space X • Lindel¨ of, if every open cover of X contains an at most countable subcover of X. • w–Lindel¨ of (= weakly Lindel¨ of), if every open cover of X has an at most countable open refinement. • vw–Lindel¨ of (= very weakly Lindel¨ of), if every open cover of X has an at most countable refinement. • s–Lindel¨ of (= strongly Lindel¨ of), if for every extension Y of X, each open cover of X in Y contains an at most countable subcover of X. Prove that: (1) s–Lindel¨ of ⇒ Lindel¨ of ⇒ w–Lindel¨ of ⇒ vw–Lindel¨ of.

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A System of Logic Ratiocinative and Inductive, Part II (The Collected Works of John Stuart Mill - Volume 08) by John Stuart Mill


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