Let $X$ be a compact topological Space. Consider a nonconstant sequence $(x_n)_{n\in \mathbb{N}}$ in $X$ such that $\{x_n:n\in \mathbb{N}\}$ is an infinite subset of $X$. Note that the said sequence can be treated as a net from $\mathbb{N}$. Obviously, this sequence has a convergent subnet (which may not be a subsequence) in $X$. Now, I am finding it difficult to symbolise this subnet. I am trying to find a notation for this subnet which does not confuse it with subsequence. Any help will be appreciated.

Thanks in advance.

  • 1
    $\begingroup$ What’s your definition of subnet ? There are several. $\endgroup$ Jun 1 at 17:38
  • $\begingroup$ I am using the en.m.wikipedia.org/wiki/…. Definition of subnet. $\endgroup$
    – pmun
    Jun 1 at 18:02
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    $\begingroup$ And see "Notes" on that page for the alternate definition. Notation will be the same, regardless of which definition is used. $\endgroup$
    – GEdgar
    Jun 1 at 18:20

So $(x_n)_{n \in \mathbb N}$ is a sequence. A subnet is like this: $J$ is a directed set, and for each $\alpha \in J$ we have an $n(\alpha) \in \mathbb N$. The subnet is then $(x_{n(\alpha)})_{\alpha \in J}$. [Some conditions are required, but they are not shown in the notation.]

This is similar to the subsequence notation $(x_{n_k})_{k \in \mathbb N}$. But easier to read without a double subscript.


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