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Let's consider a topological space that is not necessarily metrizable.

Question: I wonder what is the motivation for defining convergence of nets in a topological space? What do we gain in working with convergence of nets rather than convergence of sequences?

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By "network" do you mean "net"? –  Andrew Uzzell Dec 18 '12 at 11:50
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Properties defined by sequences (eg sequentially compact) are often different from those not defined by sequences (eg compact) especially in spaces that are not first countable. Nets give a more general framework that corresponds to the latter concept, e.g. compactness is equivalent to every net having a convergent subnet. –  Sanchez Dec 18 '12 at 11:59
    
@Sanchez If I understand correctly topological compactness (ie by open) is equivalent to compactness defined by net? How can I prove it? You have an exemple? –  Elias Dec 18 '12 at 12:06
    
@Elias, it's net, not network if I understand your question correctly. A quick google search returns this: planetmath.org/encyclopedia/… –  Sanchez Dec 18 '12 at 12:08
    
@Sanchez, thank's. I up you coment. –  Elias Dec 18 '12 at 12:10
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1 Answer

up vote 6 down vote accepted

If $X$ is a metrizable space, the topology of $X$ is completely determined by the convergent sequences: if we know which sequences in $X$ are convergent, and what their limits are, we can determine exactly which subsets of $X$ are open. This is actually true in a somewhat larger class of spaces than just metrizable spaces, but it is not true for topological spaces in general.

For example, let $X$ be an uncountable set, let $\tau_0$ be the discrete topology on $X$, and let $\tau_1$ be the co-countable topology on $X$. Then $\langle X,\tau_0\rangle$ and $\langle X,\tau_1\rangle$ are not homeomorphic, but they have exactly the same convergent sequences. If $Y$ is any uncountable subset of $X$ such that $X\setminus Y$ is also uncountable, then $Y$ is open in the discrete topology but not in the co-countable topology. And in each topology the convergent sequences are precisely the sequences that are eventually constant.

Nets are a generalization of sequences powerful enough to capture the topology of any space, not just metrizable spaces: if $\tau_0$ and $\tau_1$ are topologies on a set $X$ that have exactly the same convergent nets, then $\tau_0=\tau_1$. This, very simply, is the main motivation for looking at them.

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I just read online that filter was another proposal to generalize sequences. Do filters capture the topology of a space in the same sense? –  Sanchez Dec 18 '12 at 12:11
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@Sanchez: Yes, they do. In fact, there’s a way to convert automatically between statements about filters and statements about nets, though in any given context one may be more convenient than the other. –  Brian M. Scott Dec 18 '12 at 12:12
    
@Brian M. Scott. There is a version of the Arzela-Ascoli theorem for convergence in net? –  Elias Dec 18 '12 at 12:13
    
Thanks! Is there any (succint) reference that compares nets and filters? I would like to know more about their similarities and differences. –  Sanchez Dec 18 '12 at 12:14
    
@Sanchez: This PDF is pretty good. –  Brian M. Scott Dec 18 '12 at 12:14
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