3 11 Let X R 2 Recall that a slice set in X is one of the form S x u r v X hv

3 11 let x r 2 recall that a slice set in x is one of

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11.LetX=R2. Recall that a slice set inXis one of the formS(x,u, r) ={vX:|hv-x,ui|< r}wherex,uXandr >0.i)Verify thatG={∅} ∪ {finite intersections of slices}.is a base for a topologyτonX.ii)Prove thatτis just the usual topology!iii)Explain, using the previous question, why when you do this on2, you don’tget the usual norm topology.12.Prove that the gate sets form the base for a topology (see Exercise 4.5.1).13.True or false? Suppose thatτ1andτ2are two topologies onXwith the propertythat wheneverxαxin theτ1topology thenxαxin theτ2topology. Thenτ2τ1. Assignment 4 Assignment 4, worth 5%, is due Midday Friday 1 May. Assignment 4 consists of questions 3 and 7. 4
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