Find two different topologies on where one is strictly finer, yet the two are homeomorphic

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- Feb 24th 2009, 07:14 PMAndreametHomeomorphic finer topology
Find two different topologies on where one is strictly finer, yet the two are homeomorphic

- Feb 24th 2009, 10:27 PMaliceinwonderland
and as subspaces of R with a discrete topology.

Let and be bases for topologies and .

is strictly finer than , since for each x in Z and for each basis element containing x, we have a basis element in that contains x. The converse is not necessarily true.

Since there is a bijection between Z and Q, we have a homeomorphism between and as subspaces of R with a discrete topology.