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Let $(X,\mathcal{T}$) be a 1st countable topological space. Let $(x_\delta)_{\delta\in\Delta}$ be a net converging to $x$. Does there exist a sequence $(x_n)$ that converges to $x$ and which is a subnet of $(x_\delta)$? Any feedback is most appreciated?

Hendrik
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Certainly not. Even if the net $(x_\delta)$ has the amazing property $x_\delta = x$ for all $\delta$. Even then, it may have no subsequences at all.

GEdgar
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  • I have no idea of set theory, but is the hypothesis $|\Delta| \geq |\Bbb N|$ (in some precisely defined sense that I'm not totally sure of) what we'd need? – GPerez Apr 28 '15 at 13:59
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    An example for this is $\Delta = \omega_1$, the least uncountable ordinal. – GEdgar Apr 28 '15 at 14:01
  • In which case there do exist subnets, right? – GPerez Apr 28 '15 at 14:04
  • Thanks you for your comments, but can you tell me under what conditions will a net have a subnet which is a sequence that converges to the same limit. Also under what are the sufficient and necessary conditions for a top space to be characterized by it's convergent sequence? Thanks again I really appreciate it. – Hendrik Apr 28 '15 at 15:15
  • @Hendrik: You should ask these as new questions, instead of appending them as a comment here where very few people will see them. But first you may wish to read https://en.wikipedia.org/wiki/Sequential_space – Nate Eldredge Apr 28 '15 at 15:17
  • This depends on which definition of subnet you use. If you use the nicest one, due to Aarnes and Andenæs, every constant net has the corresponding constant sequence as a subnet and hence as a subsequence. All three are discussed in this answer. – Brian M. Scott Apr 28 '15 at 18:13