$\{x_j\}_{j\ge i}$ is a subnet of $\{x_i\}_{i}$?

Question

Let $\{x_i\}$ be a net. Then, for any $i$, we take $\{x_j\}_{j\ge i}$. Then, $\{x_j\}_{j\ge i}$ is a subnet of $\{x_j\}_{i}$.

Is the above statement correct?

I have checked wikipedia for many times. But I am not very familiar with general topology.

Answer 1

A net is a function $x:I\to X $ where $I$ is a directed set.

For $i\in I$ set $I_i = \{j\in I: j\ge i\}$ is also a directed set so the restriction $x|_{I_i}$ is a net.

To show that $x|_{I_i}$ is actually a subnet of $x$ you need to show that there is a monotone function $f:I_i\to I$ such that $(x|_{I_i})_{j}= x_{f(j)}$ and this function is final.

If you choose $f$ so that $f(i)=i$ then it is monotone. For $j \in I$ there is $k \in I_i $ such that $j \le k = f(k)$ (I'll leave you to justify this) so $f$ is final.

Hence you have your result. Note that there is no mention of topology