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Consider whether a certain type $\mathcal C$ of convergence of filters, nets or sequences in a set $X$ is topologizable. If I understand correctly, "topologizable" means there exists a topology, s.t. the given convergence is the same as the convergence wrt the topology.

If $\mathcal C$ is topologizable, is the topology induced by $\mathcal C$ on $X$ the smallest or biggest topology over $X$ s.t. $\mathcal C$ is the same as the convergence wrt the topology?

To answer the above question, I would like to consider the following one. If $\mathcal C$ is the same as the convergence wrt a topology $\tau$ over $X$, and $\tau_1$ and $\tau_2$ are two other topolgoies over $X$ s.t. $\tau_1 \subseteq \tau \subseteq \tau_2$, is $\mathcal C$ the same as the convergence wrt the topology $\tau_1$ or $\tau_2$? I think $\mathcal C$ is the same as the convergence wrt the topology $\tau_1$ but not necessarily wrt $\tau_2$, right?

Thanks and regards!

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Definitely not.

Look at a familiar case: let $\tau$ be the usual topology on $\Bbb R$, $\tau_1$ the indiscrete topology, and $\tau_2$ the discrete topology. In $\langle\Bbb R,\tau_1\rangle$ every sequence converges to every point; in $\langle\Bbb R,\tau_2\rangle$ the only convergent sequences are the eventually constant sequences, each of which converges to the point at which it is eventually constant. And in $\langle\Bbb R,\tau\rangle$ the sequence $\langle 2^{-n}:n\in\Bbb N\rangle$ is not constant converges only to $0$. The three topologies have three distinct families of convergent sequences.

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Thanks! Can a given family of convergent filters, nets or sequences only be topologizable by at most one topology? Or it may or may not? –  Tim Feb 24 '13 at 3:41
    
@Tim: Two different topologies on the same set can have the same convergent sequences (e.g., the discrete and co-countable topologies on an uncountable set). However, if two topologies on $X$ have the same convergent nets or filters, then they’re the same topology. –  Brian M. Scott Feb 24 '13 at 4:02
    
So topologizing a type of convergence has nothing to do with fineness of topologies, does it? –  Tim Feb 24 '13 at 4:09
    
@Tim: I’d not go quite that far: the finer the topology, the fewer the convergent nets/filters. But if your set of convergent nets/filters is the set of all convergent nets/filters in some topological space, then it determines a unique topology. With sequences this need not be the case. –  Brian M. Scott Feb 24 '13 at 4:19
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