General Topology 22M 132
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This 5 page Class Notes was uploaded by Virgil Wyman on Friday October 23, 2015. The Class Notes belongs to 22M 132 at University of Iowa taught by Jonathan Simon in Fall. Since its upload, it has received 28 views. For similar materials see /class/227988/22m-132-university-of-iowa in Mathematics (M) at University of Iowa.
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Date Created: 10/23/15
22M132 Fall 07 J Simon Sample Problems for Exam ll Problem 1 a Show that R with the product topology is metrizable b Show that R with the bow topology is not metrizable Problem 2 Let X be a metric space with metric d and let A be a nonempty subset of X De ne afunction dA X a R by dAz infdda a E A a Prove the function dA is continuous b Suppose A and B are disjoint closed sets in the metric space X and assume in addition that A is compact Prove there epists A gt 0 such that for all a E Ab E B dab 2 A C Give an eccample to show we cannot omit the assumption quotA is compact in part b that is give an eccample of a metric space X with disjoint closed sets A B such that E a distance A gt 0 between the sets Problem 3 Let fn be a sequence offunctions fn X a Y where X is a topological space and Y is a metric space and let f X a Y be some function a De ne the statement fn converges uniformly to f b Prove If fn converges uniformly to f and each fn is continuous then f is continuous C Prove IfX is compact and fn converges uniformly to f and each fn is continuous and surjective then f is surjective Problem 4 a De ne quotient map b Eccplain why the following function is not a quotient map here 1 is the unit circle in R2 f 0 27139 a S1 given by ft cos tsin t J Simon all rights reserved page 1 Problem 5 Suppose f X a Y is a surjective continuous function X is compact and Y is Hausdor Prove that f is a quotient map Problem 6 Suppose f X a Y is continuous 1 1 and surjective X is compact and Y is Hausdor Prove f is a homeomorphism Problem 7 Let X S1 and de ne an equivalence relation on X by saying each point is equivalent to its antipode ie 7y N i 7 39 So each equivalence class consists of eccactly two points Let Y be the quotient space X N Prove Y is homeomorphic to S1 Hint De ne afunction f Y a SL that takes each equivalence class to a single point of S1 Then use previous problems to show f is a homeomorphism Problem 8 Show that the following properties of a space X are equivalent to each other a X A U B where A and B are disjoint open sets b X A U B where A and B are disjoint closed sets C X A U B where neither of AB intersects the closure of the other Problem 9 Suppose X A1 U A2 U where each set Al is connected and Vi Al Ai1 31 0 Prove X is connected Problem 10 Suppose X is connected and f X a Y is a surjective continuous map Prove Y is connected J Simon all rights reserved page 2 Problem 11 a De ne path connected b Prove the continuous image of a path connected space is path connected C Prove IfX and Y are path connected then X gtlt Y is path connected State carefully whatever lemmas you use Problem 12 Prove IfA is a connected subset of a space X then the closure A is connected Problem 13 For this problem assume we have proven that R is connected and that nite products of connected spaces are connected The problem has two related parts Let A an E R an 0 for all but nitely many Here R has the product topology 1 Show A is connected Hint Write A as a nested union of connected sets ii Use your result in part to show that R is connected Problem 14 Prove the interval 01 is connected Problem 15 a Use Problems 9 and 14 to show that R1 with the standard topology is connected b Show that R is totally disconnected Problem 16 Suppose f Sl a R is a continuous function Prove there epists z E 1 such that fp ex Problem 17 Recall that a space X is locally path connected iffor each x E X and each neighborhood U of p there epists a neighborhood V ofz such that V Q U and V is path connected Prove IfX is connected and locally path connected then X is path connected More generally ifU is a connected open subset of a locally path connected space then U is path connected J Simon all rights reserved page 3 Problem 18 a De ne connected component b Prove that the components of any space are closed c Given an eccample to show that components do not have to be open Problem 19 a De ne locally connected b Prove that X is locally connected if an only iffor each open set U Q X each component ofU is open Problem 20 a Prove IfA is a closed subset of a compact space X then A is compact b Prove IfA is a compact subset of a Hausdor space X then A is closed in X Problem 21 a IfX is Hausdor z E X and A Q X is a compact set that does not contain x then there eccist disjoint neighborhoods U ofz and V of A b If A B are disjoint compact sets in a Hausdor space X then there eccist disjoint neighborhoods U ofA and V of B Problem 22 Prove IfX and Y are compact spaces then the product X gtlt Y is compact Problem 23 a Give an eccample of an in nite collection of closed sets in R1 that has empty intersection but such that each nite subcollection has nonempty intersection b Prove that in a compact space X any collection of closed sets with the nite intersection property has nonempty intersection Problem 24 Prove the interval 01 C R1 is compact Problem 25 Prove that a set C Q R is compact if an only ifC is closed and bounded J Simon all rights reserved page 4 Problem 26 Suppose a compact metric space X is eppressed as the union of two open sets X U U V Prove there epists a number A gt 0 such that each subset ofX having diameter lt A is contained in U or in V Remark At the risk of pointing out the obvious this is not claiming that all the small sets are in U or all the small sets are in V Some small sets end up in U others in V and some in both Saying that a set is contained in U does not prevent it from intersecting V maybe even being contained in V as well as in U If you think of a cartoon in which a small set starts inside U and moves gradually to escape from U by the time a part of the small set gets outside U the set is entirely contained in V Problem 27 a Prove IfX is compact then X is limit point compact b Give an eccample of a space that is limit point compact but not compact Problem 28 a IfX is Hausdor z E X U a neighborhood ofz such that the boundary de is compact then there epists a neighborhood V ofz such that the closure V Q U b IfX is a locally compact Hausdor space and U is a neighborhood of a point x then there epists a neighborhood V ofz such that V Q U Problem 29 Suppose X is a compact Hausdor space and A1A2 is a countable collection of closed sets each having empty interior Prove that the union Ufa An cannot be all of X Note The same proof works to show that a locally compact Hausdor space cannot be the union of countably many closed sets with empty interiors In fact what one can show is that the complement of any countable collection of closed sets with empty interiors must itself be dense in X This is enough Happy studying J Simon all rights reserved page 5
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