What kinds of nets can be used to singlehandedly guarantee continuity?

Let $$X$$ and $$Y$$ be topological spaces. Then for any directed set $$A$$ we can define nets in $$X$$ and $$Y$$ indexed by $$A$$. And a function $$f:X\rightarrow Y$$ is continuous at a point $$x$$ if and only if all $$f$$ maps all nets converging to $$x$$, indexed by all directed sets, to nets converging to $$f(x)$$.

But something stronger than that is true. Let $$A_0$$ is the set of all open sets containing $$x$$, endowed with a partial order $$\leq$$ defined by saying that $$U\leq V$$ if $$V$$ is a subset of $$U$$. Then $$f$$ is continuous at $$x$$ if and only if $$f$$ maps all nets converging to $$x$$ indexed by $$A_0$$ to nets in $$Y$$ converging to $$f(x)$$. What that means is that if $$f$$ maps all nets converging to $$x$$ indexed by $$A_0$$ to nets converging to $$f(x)$$, then $$f$$ maps all nets converging to $$x$$ indexed by all directed sets to nets in converging to $$f(x)$$.

My question is, are there any directed sets other than $$A_0$$ which have this property? The natural numbers under the standard ordering clearly do not have this property, unless the topological space is well-behaved, since a function $$f$$ mapping sequences converging to $$x$$ to sequences converging to $$f(x)$$ can fail to be continuous at $$x$$. What about the real numbers, or the ordinal numbers?

• @WilliamElliot Thanks, I fixed it. – Keshav Srinivasan Sep 16 '18 at 3:01
• It turns out you can do quite a lot with transfinite sequences (nets with an ordinal as the domain) instead of just arbitary nets. See this paper by Howes e.g. – Henno Brandsma Sep 16 '18 at 4:51