Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces. And let $Z=X\times Y$ be the product space equipped with the natural product topology $\mathcal{T}_Z$ on $Z$. Then, let $\{z_\lambda\}_{\lambda\in\Lambda}$ be a net in $Z$ such that $\{\pi_x(z_\lambda)\}_{\lambda\in\Lambda}$ and $\{\pi_y(z_\lambda)\}_{\lambda\in\Lambda}$ are nets that have cluster points $x\in X$ and $y\in Y$, respectively, where $\pi_x$ and $\pi_y$ are the natural projections from $Z$ onto $X$ and $Y$ respectively.
Now, can somebody give an explicit example of a net $\{z_\lambda\}_{\lambda\in\Lambda}$ (and topological spaces $X$, $Y$, and hence $Z$) that satisfies the above condition, but $(x,y)\in Z$ is not a cluster point of the net?
Thank you.
Just consider $\mathbb{R}\times\mathbb{R}$ and the sequenece $\{(a_n,b_n)\}_{n\in\mathbb{N}}$ where $$ a_n=\begin{cases} n&\text{if } n\text{ is odd}\\ 0& \text{else} \end{cases} \quad\text{and}\quad b_n=\begin{cases} n&\text{if } n\text{ is even}\\ 0& \text{else.} \end{cases} $$