A simpler equivalent definition of subnet

Question

I'm reading the definition of subnet from here, i.e.,

Let $E$ be a non-empty set and $(x_d)_{d\in D}$ a net in $E$. Let $(y_t)_{t\in T}$ be another net in $E$. Then $(y_t)_{t\in T}$ is called a subnet of $(x_d)_{d\in D}$ if and only if there is a function $\varphi:T \to D$ such that

$y_t = x_{\varphi (t)}$ for all $t$.

$h$ is monotonic, i.e., $\varphi (t_1) \le \varphi (t_2)$ if $t_1 \le t_2$.

$\operatorname{im} \varphi$ is cofinal in $D$, i.e., $\forall d\in D, \exists t\in T: d \le\varphi (t)$.

I realized that this definition of subnet is equivalent to

$(y_t)_{t\in T}$ is a subnet of $(x_d)_{d\in D}$ if and only if

$T$ is cofinal in $D$, i.e., $\forall d\in D, \exists t\in T: d \le t$.

$x_t = y_t$.

Let $t_1, t_2 \in T$. Then $t_1, t_2 \in D$. There is $d \in D$ such that $t_1 \le d$ and $t_2 \le d$. Because $T$ is cofinal in $D$, there is $t_3 \in T$ such that $d \le t_3$. This implies $t_1 \le t_3$ and $t_2 \le t_3$. Hence $T$ is indeed a directed set. To get the first construction, we set the canonical injection from $T$ to $D$ as $\varphi$.

It follows that $(x_d)_{d\in T}$ is a subnet of $(x_d)_{d\in D}$ if and only if $T$ is cofinal in $D$. Could you verify if my understanding is correct?

@AndréCaldas I don't get your point. Could you explain more? — Analyst, Jan 28 '22 at 15:19
You are assuming that $T \subset D$. This assumption is not correct. — André Caldas, Jan 28 '22 at 15:20
To expand on @AndréCaldas 's comment, there are uncountable subnets of sequences. For instance $\phi:{\mathbb R}{\ge 1} \to {\mathbb N}$ defined by $\phi(r) = \lfloor r \rfloor$ gives a subnet of $(x_n){n\in \mathbb N}$: $$ (x_{\lfloor r \rfloor}){r \in {\mathbb R}{\ge 1}} $$ — Jamie Radcliffe, Jan 28 '22 at 15:35
The answer at https://math.stackexchange.com/q/2150490/25795 might be helpful. — Jamie Radcliffe, Jan 28 '22 at 15:49

score 3 · Accepted Answer · answered Jan 29 '22 at 17:37

Let me begin with a comment concerning the definition of "subnet". In my opinion one should require the function $\varphi$ to be a constitutive part of a subnet. That is, I would prefer to define a subnet as a pair $((y_t)_{t \in T}, \varphi)$ with the appropriate properties. The mere existence of a function $\varphi$ does not tell you which $x_d$ you have to associate to $y_t$; there may be many functions $\varphi$ and each of them gives you a different assocation. But I admit that this is a matter of taste and everybody is free to give the definition he thinks to be most adequate.

Your "simplified definition" is obviously less general than Willard's. You only consider cofinal subsets $T \subset D$ and $\varphi =$ inclusion function $T \hookrightarrow D$. Thus your definition is not equivalent to Willard's, you get much less subnets.

The essential question is: Does it matter? Is your simplified concept adequate for all purposes?

Unfortunately the answer is "no". In the article linked in your question you see that Willard's concept allows to prove

A net has a cluster point $y$ if and only if it has a subnet that converges to $y$.

This is not true for your concept. See Subnet vs Cofinal subnet.

A simpler equivalent definition of subnet

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