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by: Addison Beer
Addison Beer
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Judith Arms

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Judith Arms
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This 2 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 544 at University of Washington taught by Judith Arms in Fall. Since its upload, it has received 51 views. For similar materials see /class/192062/math-544-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15
Math 544 Examples October 2007 1 This document gives two kinds of examples Counterexamples for quotient spaces appear on this page and distinguishing examples for various combinations of types of connectedness are discussed on p 2 I Counterexamples for quotient spaces The quotient of a second countable space need not be second countable or even rst countable For an example let X R with its usual topology and let Y be the result of collapsing the integers to a point That is Y RZ but in the sense of Z being an equivalence class the only one that isn7t a singleton and not in the sense of the coset space for Z as a subgroup Let p be the quotient map from X to Y If y pZ and Vn is a family of neighborhoods of y proposed as a neighborhood basis for Y at y we can use a Cantor Diagonalization77 argument to construct a neighborhood W of y that doesn7t contain Vn for any n This shows that no such Vn can be a neighborhood basis for Y at y so Y is not rst countable and thus not second countable either To construct the set W let 6 be a positive number no larger than 14 such that n 7 6n2n EnQ C 1040 De ne 6n EnQ for positive integers n and 6 14 for all other integers The set U Um ion2n62 nEZ is a saturated open set containing Z so pW is a neighborhood ofy By construction no Vn is contained in W The condition in Proposition 360 that the quotient map be open cannot be omitted A counterexample is given in Munkres7s book Topology p 145 Exercise 6 The K topology used in the example is de ned on p 82 of the reference Note that the inverse image of the diagonal mentioned in part b of the exercise is exactly the set 9 in Proposition 360 The product of quotient maps may not be a quotient map Thus the conclusion of Lemma 459 is not trivial Two counterexamples are given in Munkres Example 7 on pp 143 144 shows that the product of a quotient map with the identity may not be a quotient map The example in Exercise 6 p 145 shows that a product of a quotient map with itself may not be a quotient map Math 544 Examples October 2007 2 II Distinguishing examples for connectedness For the following discussion we use the abbreviations C for connectedconnectedness P for path and L for locally Thus LPC means locally path connected We have three results relating the four kinds of connectedness C LC PC and LPG By Theorem 412 PC implies C and as an immediate corollary LPC implies LC see midpage 88 By Proposition 423c LPG and PC together imply C Given these three implications only eight combinations of the properties are possible and all in fact occur as the following examples illustrate H F 9 7 U 03 5 8 The topologist7s sine curve TSC Example 414 is C but not LC PC or LPC The disjoint union of two copies of TSC in not C LC PC or LPC If we add the points L 1 6 R210 3 x S 1 to TSC the resulting space is C and PC but not LC or LPC Let X be an in nite set with the nite complement topology This is the rst topology in Problem 2 2 Then X is C and LC but not PC or LPC 1 consulted Counterm amples m Topology by Steen and Seebach 1st ed 1970 to identify this example The cone over X is C LC and PC but not LPC Thanks to Professor Scott Osborne for suggesting this method to add only PC to an example that is C and LC The disjoint union of two copies of X gives a space that is LC but not C PC or LPC Any manifold with two or more components is LC and LPG but not C or PC A connected manifold has all four connectedness properties C LC PC and LPG Counterecoamples m Topology discusses stronger connectedness conditions hypercon nectedness77 and ultraconnectedness Also discussed is a variant of path connectedness requiring the path to be injective These variations aren7t widely used I mention them only as a curiosity IfI have the time later I hope to add a diagram to these notes summarizing the discussion of connectedness above


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