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530 Hwk 10-11 - Math 530 Hwk It From Munkres purse 1 1‘3...

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Unformatted text preview: Math 530 Hwk It} From Munkres : purse 1 1‘3. problems 1.2.3.5‘7 .1“. i. Let I and T' be topologies on a set X, and let id : <X.’t’> a <X,T'> be the identity map. (a) Prove that id is continuous ill I is stronger than I'. (b) Prove that id is open iii id is closed it? I’ is stronger than “C. Remark : this exercise is a source for basic counterexamples involving continuity versus openness or closedness. 2. Let Y be a set and let <X.T> be a topological space. Suppose t': X —> Y is a function. Prove that t" : {E i: Y : Hill) E 1'} is a topology and that n'iorem-‘er it is the strongest topology on Y such that l' is continuous. 3. Let CtXR) denote the continuous real valued functions on the top space <X,I>. (a) Let 1:ka be the weak topology on X induced by C(X.R). Prove that Tweak: T. (bl Prove that Twat. : 1' ill whenever a e X and A is a nonempty closed set disjoint from a. there exists t' E C‘iXRi with fix) 6E HA) {Necessity is not trivial and involves operations with real valued functions; remember that a basic open set in Tweak is a finite intersection ol‘ inverse images ot'opcn sets ). , . . i . . . DO 4. Let {dump : l 6 At} be a countable lamtly ol’ topological spaces. Prove that i=1 with the product topology is second countable il'and only il'each Xi is second countable. 5. Is a product ofdiscrcie spaces discrete? 6. Let {<X1.di> : i c 2+] be a countable family of metric spaces. where for ea"h i, . ' __ eta ; DO dtamtk‘i} E l. Dcltnc p: i=1)“ >< i=1 X} by pt<xi.>.<_v;>i 2 Y in 3'l - ddspyil. £— l:l Prove that p is a metric whose open sets are those oi“ the product topology. Math 530 llwk I] From Munkres page 152. problems I773) page [57. problems l.3.lO.ll 1. Use continuous l'unctions. explain why {(x.y) : x3 + 2y2 2 8] in the plane is connected. 2. Explain why ltx._vi: y ; .\ + l or y 3: —x + l is connected. using continuous functions, and Problem 3 on Page IS] on unions ot'connected sets. ...
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