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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 Winter '10
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 Topology

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