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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 Spring '08
 Ogle,C
 Topology

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