I just learned about nets, but I'm confused about when to use them. More precisely when combined with products.
Let $(X_i)_{i\in I}$ be a family of topological spaces and consider $X = \prod_{i \in I} X_i$ equipped with the product topology induced from the family $(X_i)_{i\in I}$.
I read on wiki that a net $(y_\alpha)_{\alpha \in A} $ in $X$ converges to $y \in X$ if and only if $\pi_i(y_\alpha) \to \pi_i(y)$ for all $i \in I$, where $\pi_i$ are the projection maps.
So my first question is: If the $X_i$ are first countable or sequential, can I use sequences instead of nets, or this depends on the set $I$?
The particular case I'm interested in is $X= [0,1]^{\mathcal{P}(Z)}$ where $Z$ is a given set, where the topology in $[0,1]$ is the induced by the standard topology in $\mathbb{R}$.
In this case, to check convergence of a net, is it enough to work with sequences?
If the answer is no and I still need to work with nets, does the following property of sequences hold for nets? ($X$ is Hausdorff, so if the limit of a net exists, is unique)
If $\lim y_n = y$ and $\lim z_n = z$ then
$\lim (y_n + z_n) = y + z$
Thank you in advance and I hope my questions are clear.