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