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From this question Homeomorphism that maps a closed set to an open set? the answer shows that {0},{1} are homeomorphic in the subspace topology. I feel like I have a problem understanding "two sets are homeomorphic with respect to the subspace topology", can someone explain this to me? It will be great if you can provide a few more example about homeomorphism in other topological spaces.

Thanks.

p.s I know what "homeomorphic" and "the subspace topology" mean, but I only did homeomorphism exercise in usual topology space, so I'm a little confused about what will "homeomorphic" look like in the subspace topology.

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  • $\begingroup$ Maybe you can help us by taking apart what exactly it is that you don't understand. Do you know what homeomorphic means? Do you know what "the subspace topology" means? $\endgroup$ – Omnomnomnom Oct 10 '16 at 18:27
  • $\begingroup$ Yes I know what "homeomorphic" and "the subspace topology" mean, but I'm new to them, so it is hard to understand what does "two sets are homeomorphic with respect to the subspace topology" mean. Thanks for the note! $\endgroup$ – Sandy.Davidson Oct 10 '16 at 18:29
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The example is not very instructive. The statement means that if you take the sets {0} and {1} as subspaces, the subspace topology (open sets = intersections of open sets with subspace) on each gives you that each is both open and closed. So of course the map from one to the other is a homeomorphism in that context. But it's cheating to change the topology.

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