Homeomorphism between sets

Question

Let $$ Y = \left\{ \frac{1}{2} + \frac{1}{n} \mid n \in \mathbb{N}\right\}\cup\left\{ \frac{1}{2} - \frac{1}{n} \mid n \in \mathbb{N}\right\}$$ and $$ X = \left\{ \frac{1}{n} \mid n \in \mathbb{N}\right\} $$ be subspaces of the Euclidean space $ \mathbb{R} $. Are $X$ and $Y$ homeomorphic (i.e., does there exist a continuous bijection between them whose inverse function is continuous)? I tried to construct a bijection, but, every time I try, I can't get it to be continuous. So I would assume that they are not homeomorphic.

Answer 1

Your topologies are discrete, so every map is continuous. Therefore every bijection between $\mathbb{Z}^*$ and $\mathbb{N}$ will do.