# Homeomorphic finer topology

• February 24th 2009, 07:14 PM
Andreamet
Homeomorphic finer topology
Find two different topologies on $\mathbb{R}$ where one is strictly finer, yet the two are homeomorphic
• February 24th 2009, 10:27 PM
aliceinwonderland
Quote:

Originally Posted by Andreamet
Find two different topologies on $\mathbb{R}$ where one is strictly finer, yet the two are homeomorphic

$(Z, T_1)$ and $(Q, T_2)$ as subspaces of R with a discrete topology.

Let $B_1$ and $B_2$ be bases for topologies $T_1$ and $T_2$.

$T_2$ is strictly finer than $T_1$, since for each x in Z and for each basis element $\{x\} \in B_1$ containing x, we have a basis element $\{x\}$ in $B_2$ that contains x. The converse is not necessarily true.

Since there is a bijection between Z and Q, we have a homeomorphism between $(Z, T_1)$ and $(Q, T_2)$ as subspaces of R with a discrete topology.