Recall that a topological space is called Fréchet–Urysohn if every convergent net contains (as a set) a sequence, which is convergent to the same limit. I want to refine this property as follows.
Let $X$ be a Hausdorff Fréchet–Urysohn space, and let $(x_i)_{i\in I}$ be a net in $X$, which converges to $x$.
Can we find an increasing sequence $(i_n)\subset I$ of indices such that $x_{i_n}\to x$?
Let $\{j_{n}\}\subset I$. Can we find a sequence $(i_n)\subset I$ such that $x_{i_n}\to x$ and $i_n\ge j_n$, for every $n$?
