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hmw8.1 - Math 655 — Homework fl 1 Suppose that X and Y...

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Unformatted text preview: Math. 655 — Homework # fl 1) Suppose that X and Y are Tg—spaces. Show that X In: F is also T3. (Hint: First reformulate the T3 condition in an analogous way to the way we reformulated T4 in the proof of Uryohn's Lemma.) 2} Let {(Xm Paflaea be an uncountable collection of metric spaces, each containing at least two points. Show that n XE is not first countable, hence not metrizable. [X is first countable if for each s: E X we can find a countable collection of open sets {Ufl}f=l containing 3:, such that any open set containing 3: includes at least one of the sets UH.) 3) Let {[3}, Pangaea be as above. Can you find a metric on n XE... such that [a] the projection maps 1—] X“ —} X3 (b) the injection maps Xfl —} n XE. are simultaneously continuous? 4) Show that the Cantor ternary set [“11] — [(1/33/54) Ullfgflfgl U WHEN) U - - -l l ...
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