General Topology/Metric spaces

This definition is justified:

Now subsets of metric spaces are again metric spaces (as can be easily verified), and the topology induced by the restriction of the metric of the larger space is the subspace topology:

As usual, by choosing ${\delta }_x$, ${\delta }_y$ maximal by choosing the union of all the open balls satisfying the condition, one may avoid the axiom of choice.

Exercises

1. Let $M$ be a compact topological space, whose topology is induced by two metrics $d$ and $d^{\prime }$. Prove that for every $ϵ>0$, there exists $\delta >0$ such that $d(x,y)<\delta$ implies $d^{\prime }(x,y)<ϵ$ (and then by symmetry also the other way round).