How to explain "mathematically" why not every net is a sequence?

Question

A past question ask to "Give an example of a net that is not a seqence and explain why mathematically".

I know that the function $f:((0,1),\leq)\rightarrow\mathbb R$ is a net (since $((0,1),\leq)$ is a directed set) but not a sequence because $((0,1),\leq)$ is not $\mathbb N$. But how do I explain this mathematicaly. Do I have to prove that $((0,1),\leq)$ is a directed set and $((0,1),\leq)$ is not $\mathbb N$? How to do that?

In a sequence every tail set omits at most finitely many points. Not so in your $f$. A clear distinguishing feature IMO. The question is vague anyway. — Henno Brandsma, Nov 07 '21 at 17:23

score 3 · Answer 1 · answered Nov 07 '21 at 13:39

3

No. You have to prove that $\bigl((0,1),\leqslant\bigr)$ and $(\Bbb N,\leqslant)$ are not order-isomorphic. In other words, that there is no bijection $\beta\colon(0,1)\longrightarrow\Bbb N$ such that $\beta(x)\leqslant\beta(y)\iff x\leqslant y$. That's easy, since there is no bijection at all from $(0,1)$ onto $\Bbb N$.

answered Nov 07 '21 at 13:39

José Carlos Santos

440,053

So do I just say that there is no bijection from $(0,1)$ onto $\Bbb N$ or do I have to prove it? – gbd Nov 07 '21 at 13:41
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Do I have to prove it? In a paper, you would site a textbook proof that $(0,1)$ is uncountable. In a class where that uncountability has not been proved? Ask the instructor. – GEdgar Nov 07 '21 at 14:09
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More interesting examples are nets that are not even transfinite sequences, because then the result doesn't rely on a trivial cardinality observation (but +1 because a "trivial cardinality observation" is a simple answer to the OP's actual question). See is a net stronger than a transfinite sequence for characterizing topology? and convergence of nets vs. convergence of really long sequences and my comments to Intuition behind nets. – Dave L. Renfro Nov 07 '21 at 14:55

How to explain "mathematically" why not every net is a sequence?

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