Basic Analysis I — Real Analysis · Problem 7.2.9

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · open-and-closed-sets

Problem

Let XX be a set and d,d′d, d' be two metrics on XX. Suppose there exists an α>0\alpha > 0 and β>0\beta > 0 such that α d(x,y)≤d′(x,y)≤β d(x,y)\alpha\, d(x,y) \leq d'(x,y) \leq \beta\, d(x,y) for all x,y∈Xx, y \in X. Show that UU is open in (X,d)(X,d) if and only if UU is open in (X,d′)(X,d'). That is, the topologies of (X,d)(X,d) and (X,d′)(X,d') are the same.

Related theory

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