General Topology – Equivalent Metrics and Convergent Sequences

general-topologymetric-spaces

I'm currently reading the book Introduction to Topology by Gamelin.
There is a problem on the first chapter that I could not figure out. Could anyone give me some hints please?

Two metrics $d,p$ on $X$ are equivalent if they determine the same open subsets. Show that two metrics $d,p$ on $X$ are equivalent if and only if the convergent sequences in $(X,d)$ are the same as the convergent sequences in $(X,p)$.

Thank you very much.

Best Answer

Hints:

  1. A subset of a metric space is open if and only if its complement is closed.
  2. A subset $A$ of a metric space is closed if and only if for every convergent sequence $a_n \to x$ in $X$ with $a_n \in A$ for all $n$ we also have $x \in A$.
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