Accumulation point of a net (topological spaces)

[mahler1]

Homework Statement .

If $(x_{\alpha})_{\alpha \in \Lambda}$ is a net, we say that $x \in X$ is an accumulation point of the net if and only if for evey $A \in \mathcal F_x$, the set $\{\alpha \in \Lambda : x_{\alpha} \in A\}$ is cofinal in $Lambda$. Prove that $x$ is an accumulation point of the net if and only if there is a subnet of $(x_{\alpha})_{\alpha \in \Lambda}$ that converges to $x$.

The attempt at a solution

I am having some difficulty proving the two implications.

→ If the set $A=\{\alpha \in \Lambda : x_{\alpha} \in A\}$ is cofinal in $\Lambda$, then for every $\alpha \in \Lambda$ there exists $\beta \in A$ such that $\beta \geq \alpha$. I want to construct a subnet that converges to $x$, first I thought of constructing a subnet $(y_{\beta_A})_{\beta_{A} \in \Lambda}$ defined as $y_{\beta_A}$ an element of the set $A=\{\alpha \in \Lambda : x_{\alpha} \in A\}$ which is nonempty. It is clear that it "converges" to $x$, but I am not so sure if this is a subnet.

For the other implication I have no idea how to start, I would appreciate suggestions.

 

[Fredrik]

I'm not familiar with these definitions, but the problem interests me since I was wondering yesterday how a subnet is defined. I had to look up "cofinal" at Wikipedia. The sentence

Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”.​

makes me believe that a set $B\subseteq\Lambda$ is cofinal with respect to the partial order on $\Lambda$ if and only if $(x_\alpha)_{\alpha\in B}$ is a subnet. (Isn't this the definition of "subnet"?). It seems to me that this observation makes the problem rather trivial? Do you disagree? (Did I perhaps guess the definition of "subnet" wrong?)

Edit I see now that I made a blunder. The problem is not trivial, even if I have understood the definitions correctly. I will think about this some more, and edit this post or make a new one when I'm done.

 

[Fredrik]

I gave it a shot, but I haven't gotten any further than you did. When I say "open nhood", I always mean "open neighborhood of x". For all open nhoods F, I will use the notation $\Lambda_F=\{\alpha\in\Lambda|x_\alpha\in F\}$.

Suppose that x is an accumulation point. For all open nhoods F, $\Lambda_F$ is cofinal. Since cofinal sets are non-empty, this implies that we can define a set S that consists of one element from each $\Lambda_F$. For each open nhood F, denote the element chosen from $\Lambda_F$ by $\alpha_F$.

For each $\alpha\in S$, there's an open nhood F such that $\alpha=\alpha_F\in\Lambda_F$ and $x_\alpha\in F$. The map $F\mapsto \alpha_F$ is a net that seems to converge to x, but that's not the net we need to converge. I think we're interested in the map $\alpha_F\to x_{\alpha_F}$ with domain S, i.e. the net $(x_\alpha)_{\alpha\in S}$. Can we prove that S is cofinal, and that the convergence of the former net implies the convergence of the latter?

I haven't made a really serious attempt to complete this approach, so I don't know if I'm on the right track.

The other implication seemed like a straightforward application of the definitions at first, but at the end I got stuck attempting to prove that a certain set is cofinal. I'll take another look tomorrow, if you haven't figured these things out for yourself by then.

 

[Fredrik]

I think I have proved the other implication, i.e. that if $(x_\alpha)$ has a subnet that converges to x, then $\Lambda_E$ is cofinal for all open nhoods E. It was helpful to prove this as a warm-up first: Let I be a directed set. Let J be an arbitrary subset of I. Let $i_0\in I$ be arbitrary.
(a) If J is cofinal, then J is a directed set.
(b) $\{i\in I|i\geq i_0\}$ is cofinal.

Since this is the homework forum, I can't just type up the proof. At this point I can only encourage you to do the above as an exercise, and then apply the definitions carefully. What does it mean to say that $(x_\alpha)$ has a subnet that converges to x? Write that down and then try to prove that for all open neighborhoods E, $\Lambda_E$ is cofinal.