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I understand that in $(\Bbb R ,d)$, any open interval $(a,b)$ is topologically equivalent to $\Bbb R$. But here the topology is unspecific, so I have no idea how to proceed.

Could anybody help me? Thanks!

Shi Ning Sun
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    $(a,b)$ should have the subspace topology. Otherwise, we can equip it with, say, the discrete topology and of course it is not homeomorphic to $\mathbb R$. – Quang Hoang Sep 16 '15 at 14:55
  • @QuangHoang But can the topology of the real line be unspecific? In that case I'm unable to associate it with the metric space $(\Bbb R, d)$ – Shi Ning Sun Sep 16 '15 at 15:11
  • When you say real line, it means the usual metric topology. – Quang Hoang Sep 16 '15 at 15:12
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    I think that the standard topology on ${\mathbb R}$ is assumed, because otherwise it is easy to give counterexamples: Adopt the usual topology on $[a,b]$, and declare all singletons ${x}$ with $x<a$ or $x>b$ as open. – Christian Blatter Sep 16 '15 at 15:13

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