Moderate# Maps and Convergence of Nets

TOPO-5X1BTU

Let $(X,\mathcal{T})$ be a topological space, $A$ a directed set, and $a:A\to X$ a net in $X$. Assume in addition that for every point $x\in X$ and every open neighborhood $U$ of $x$, one can find a function $f_{x,U}:X\to \mathbb{R}$, continuous at $x$, and satisfying $f_{x,U}(x)=1$, and $f_{x,U}(y)=0$, for all $ y\not\in U_x$.

Find the four *mutually equivalent* statements in the following answer choices.