How can I think about a net in topology?

Question

I want to ask if someone could give me some intuition about nets in topology. I know the following definition:

Let $(M,\tau)$ be a topological space an $(I,\leq)$ a directed set. This means that $I$ satisfies the following properties:

$\forall i\in I; i\leq i$

$\forall i,j,k\in I, i\leq j, \,\text{and}\,\, j\leq k \Rightarrow i\leq k$

$\forall i,j\in I \,\,\exists k\in I \,\,s.t. \,\, i\leq k\,\,\text{and}\,\, j\leq k$

Then A map $$a:(I,\leq)\rightarrow M; i\mapsto a_i$$ is called a net in M indexed by I.

But are these $a_i$'s arbitrary elements of $M$, as in a sequence i.e. we construct something similar as a sequence in $\mathbb{R}$ but in a topological space?

Moreover, when we have defined the notion of nets we can talk about the sequential closure. Is this the set of all limit points of convergent nets or only the set of specific limit points of convergent nets?

Thank you for your help.

No, "sequential closure" involves only sequences. In a general topological space, it may differ from "closure". Note that closure is the same as "net closure", so there is no need for a special term like "net closure". — GEdgar, Jan 01 '22 at 15:02
Regarding sequences and nets, see Intuition behind nets AND Sequential properties of topological spaces. Regarding transfinite sequences and nets, see is a net stronger than a transfinite sequence for characterizing topology? AND convergence of nets vs. convergence of really long sequences. — Dave L. Renfro, Jan 01 '22 at 18:14

How can I think about a net in topology?

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