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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 halfinfinite 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 halfinfinite 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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 Fall '09
 ZoltánSzabó
 Topology, @, Topological space, standard topology, Professor Zoltan Szabo, Rik Sengupta

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