325 HW 01 - MAT 325 Topology Professor Zoltan Szabo Problem...

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Unformatted text preview: MAT 325: Topology Professor Zoltan Szabo Problem Set 1 Rik Sengupta rsengupt @ princeton.edu February 7, 2010 1. Munkres, p. 83, problem 7 Consider the following topologies on R : T 1 = the standard topology, T 2 = the topology of R K T 3 = the finite complement topology, T 4 = the upper limit topology, having all sets ( a,b ] as basis, T 5 = the topology have all sets (-∞ ,a ) = { x : x < a } as basis. Determine, for each of these topologies, which of the others it contains. Solution. Note first of all, that if a ∈ R , then (-∞ ,a ) = ∪ b<a ( b,a ) is clearly open in the standard topology, and similary, so is ( a, ∞ ). We will prove all claims one by one. • T 3 ⊂ T 1 Any subset of R with a finite complement is the union of the open intervals between its finitely many complement points, and the half-infinite open intervals at both ends. Now, the open intervals between the complement points are of the form ( a,b ) and are by definition open in the standard topology. Furthermore, the half-infinite open intervals at the ends are also open in the standard topology, as we showed just now. Therefore, any set with a finite complement is in the standard topology. So T 1 is finer than T 3 . However, the set (0 , 1) ⊂ R is open in the standard topology, but does not have a finite complement. So, T 1 is strictly finer than T 3 . • T 5 ⊂ T 1 All the basis sets of T 5 are open in the standard topology, as we showed just now, and so any unions are open in the standard topology as well. Therefore, the whole topology T 5 is contained in T 1 . But in T 5 , any union of the basis elements is unbounded below. So it does not, for instance, contain (0 , 1), because of its lower bound. However, (0 , 1) is obviously open in the standard topology. Hence, T 1 is strictly finer than T 5 . 1 • T 1 ⊂ T 2 This inclusion is obvious. Definitionally, T 2 contains the whole basis { ( a,b ) : a,b ∈ R } of T 1 , and so it contains the whole topology T 1 . However, any element like (- 1 , 1)- K [where K = { 1 n : n ∈ N } ] is not open in the standard topology, though it is obviously in the R K topology. Therefore, T 2 is strictly finer than...
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This note was uploaded on 05/07/2010 for the course MAT 325 taught by Professor Zoltánszabó during the Fall '09 term at Princeton.

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325 HW 01 - MAT 325 Topology Professor Zoltan Szabo Problem...

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