# topology via converging nets

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## Mathematics Subject Classification

### How to specify a "sequential" topology via its convergent se...

I found the article on "topology via net convergence" very interesting, but it also made me think of something: is it possible to do the same but restricting our attention only to sequences? For it happens often in Analysis that a certain topological structure is constructed by giving a definition of what it means for a sequence to converge (for example, in the definition of the topology in a space of distributions), and in such constructions it is rarely proven that the definition is in fact consistent with some actual topology, from which the convergence relation comes from...

Being more specific, my question is this: Â¿is there a set of conditions that are both necessary and sufficient for a relation pairing sequences with points in a space to be consistent with a topology in wich the relation is "convergence"?

### How to specify a "sequential" topology via its convergent se...

I found the article on "topology via net convergence" very interesting, but it also made me think of something: is it possible to do the same but restricting our attention only to sequences? For it happens often in Analysis that a certain topological structure is constructed by giving a definition of what it means for a sequence to converge (for example, in the definition of the topology in a space of distributions), and in such constructions it is rarely proven that the definition is in fact consistent with some actual topology, from which the convergence relation comes from...

Being more specific, my question is this: is there a set of conditions that are both necessary and sufficient for a relation pairing sequences with points in a space to be consistent with a topology in wich the relation is "convergence"?

### Re: How to specify a "sequential" topology via its convergen...

Yes, there is such a thing as a sequential convergence space
where the convergence structure is specified through convergent
sequences rather than open sets. As for the axioms defining
such a structure, I will post them later this evening because,
right now, I have to go somewhere.