Let $\mathbb{E}$ denote the set of all even integers and $\mathbb{Z}$ be the set of all integers. $T_\mathbb{E} = \{ U \subset \mathbb{R} \ | \ U = \emptyset \text{ or } \mathbb{E} \subset U\}$ and $T_\mathbb{Z} = \{U \subset \mathbb{R} \ | \ U = \emptyset \text{ or } \mathbb{Z} \subset U\}$ are topologies on $\mathbb{R}$. How would I go about finding a homeomorphism between these sets? $f(X) = X\cup (X+1)$ seemed to work in the forward direction, but the inverse doesn't behave well. Any hints would be appreciated.
Homeomorphism between two topologies, one finer than the other
general-topology
Best Answer
Hint: If you define$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&2x,\end{array}$$then $f(\mathbb{Z})=\mathbb E$.