Visualizing nets in a topological space

Question

What is the best way one can visualize the concept of a net or a subnet in a given topological space $X$? What is the intuition that makes sence when thinking about the definitions of convergence and cluster points of a net? I understand that the convergence concept derermines the topological structure of $X$ by looking at thd theorems but I can't see how is this working and what is really going on beneath. I think once I can develop an intuituon around nets, subnets and cluster points, I could see everything more clear.

Answer 1

I have though about the question and now I think I have developed a good intuition about the concept. Indeed as @HennoBrandsma has mentioned in the comments, intuition comes from studying how they interact in the structure of a proof. The following is what I realized earlier today when I was trying to prove these two statements:

A point $p$ in a topological space is a cluster point(points that each open set $U$ containing $p$ contains all the terms of the net eventually) of a net $\{x_{\alpha}\}_{\alpha \in I}$ if and only if $\{x_{\alpha}\}_{\alpha \in I}$ has a sub-net $\{x_{\alpha_{\beta}}\}_{\beta \in J}$ such that $\{x_{\alpha_{\beta}}\}_{\beta \in J} \longrightarrow p$.

Let $X$ be a topological Space, then $X$ is Compact if and only if every net in $X$ has a sub-net converging to a point.

While proving the first statement I was first choosing terms like $x_{\alpha_{U}}$ to be any point inside an open set $U$ containing $p$ in vain. I was trying hard to prove that this should be a sub-net while all along it wasn't. I believe one important fact is that you understand that in case of a net, when you are trying to find a sub-net(which is basically finding a sub-directed set in the original directed set) choosing infinitely many indices from the index set $I$ isn't going to automatically give you a sub-net! This is important and it is a big difference with sequences. I believe my confusion was coming from this illusion that I had.

To show why this is actually the case Let's look at this picture:

The concept of a directed set is the generalization of $\mathbb{N}$. Indeed I think we can add a concept of infinity point for directed sets which I define it as follows:

Define a mere symbol $\infty$ and add it to the directed set. you can still define the partial ordering by saying $\forall x \in D \quad x < \infty$ (here, $D$ is the directed set). The condition that $\forall x, y \in D - \{\infty\} \quad \exists z \in D - \{\infty\} \quad z > x, z > y$ simply means that the points on $D$ are moving towards this $\infty$ point as they get bigger.

This pic also shows why just choosing infinitely many members do not end up as a good sub-index! you may really end up with another directed set (as I did) but this subset's infinity may not be the same as before. For example the rectangular part in the picture could be itself a manifold and has infinitely many points but all of them be on a certain level in the partial ordering so that they are not even comparable. This is why in the definition of sub-net you are required to choose indices $\alpha_{\beta}$ such that for each $\alpha \in I$, there is a $\beta_0$ such that $\beta > \beta_0 \Longrightarrow \alpha_{\beta} > \alpha$.

Before writing this answer, I also thought of this very last condition as follows. I add another symbolic point $-\infty$ and think of it as a source, for example a faucet containing some water. As the water passes through $\mathbb{N}$ or $D$ it naturally makes some points wet. But it never stops on some specific region (like the rectangular aria that could be a manifold) since the source of gravity ($\infty$ symboll) always pulls it down. This naturally gives you a sub-net.