Further Analysis | Part IB, 2004

(i) Let XX be the set of all infinite sequences (ϵ1,ϵ2,)\left(\epsilon_{1}, \epsilon_{2}, \ldots\right) such that ϵi{0,1}\epsilon_{i} \in\{0,1\} for all ii. Let τ\tau be the collection of all subsets YXY \subset X such that, for every (ϵ1,ϵ2,)Y\left(\epsilon_{1}, \epsilon_{2}, \ldots\right) \in Y there exists nn such that (η1,η2,)Y\left(\eta_{1}, \eta_{2}, \ldots\right) \in Y whenever η1=ϵ1,η2=ϵ2,,ηn=ϵn\eta_{1}=\epsilon_{1}, \eta_{2}=\epsilon_{2}, \ldots, \eta_{n}=\epsilon_{n}. Prove that τ\tau is a topology on XX.

(ii) Let a distance dd be defined on XX by

d((ϵ1,ϵ2,),(η1,η2,))=n=12nϵnηnd\left(\left(\epsilon_{1}, \epsilon_{2}, \ldots\right),\left(\eta_{1}, \eta_{2}, \ldots\right)\right)=\sum_{n=1}^{\infty} 2^{-n}\left|\epsilon_{n}-\eta_{n}\right|

Prove that dd is a metric and that the topology arising from dd is the same as τ\tau.

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