Definition of subsequence of a net

Question

I'm self studying Topology from Michael Gemignani's Elementary Topology. The author asks the following question (Exercise 2 on page 127):

Suppose $X,D$ is a metric space and $\{ s_i \} , i \in I$, is a net in $X$ and suppose that $s_i \to x$. Prove that a subsequence of $\{ s_i \} , i \in I$, converges to $x$.

I'm not sure what the author means by the subsequence of a net. The author defines subnets. Is subsequence of a net is a subnet indexed by the directed set $\mathbb{N}$? If so, we're done since we know that every subnet will converge to the same point where the net does.

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Here's Gemignani's definition of subnet:

Let $\{ s_i \} , i \in I$ be a net in $X$. Let $J$ be a directed set and $k: J \to I$ such that

• if $j \le j'$ then $k(j) \le k(j')$

• if $i, i' \in I$, then there is $j \in J$ such that $i \le k(j)$ and $i' \le k(j)$.

The composition $s \circ k$ is said to be subnet of the net $\{ s_i \} , i \in I$.

Answer 1

Is subsequence of a net is a subnet indexed by the directed set $\mathbb{N}$?

Yes. The same definition is given in the book.

If so, we're done since we know that every subnet will converge to the same point where the net does.

That depends. If the author literally meant "any subsequence of $(s_i)$ converges to $x$" then this is trivially true, as you've noted. Note that if $(s_i)$ has no subsequences (which can happen) then this is vacuously true.

However this interpretation doesn't bring anything new to the table. It seems that the author meant "$(s_i)$ has a subsequence convergent to $x$". Which unfortunately is not true in general. Recall the definition of a subnet:

Definition: For a net $f:I\to X$ a subnet of $f$ is a net $g:J\to X$ together with a monotone, cofinal function $h:J\to I$ such that $f\circ h=g$.

Cofinal here means that the image of $h$ is cofinal with $I$.

And thus the second interpretation of the statement is false. Simply because there are ordinals of uncountable cofinality. I.e. there is an ordinal $\lambda$ such that no countable subset of $\lambda$ is cofinal with $\lambda$. And so a constant net $t:\lambda\to X$, $t(i)=x$ is obviously convergent, but it has no countable subnet. Note that a constant sequence $x_n=x$ is not a subnet of $t$ - the cofinality axiom is not satisfied.

I think what the author wanted to say is that in metric spaces nets can be replaced with sequences. Meaning if $s:I\to X$ is a convergent net then there is a sequence $t:\mathbb{N}\to X$ convergent to the same limit and such that $im(t)\subseteq im(s)$. But such sequence need not be a subnet. Being a subnet is a stronger condition. For metric spaces pretty much any property requiring a net, can be stated with sequences instead.