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At the beginning the concept of limit is introduced for sequences thanks to the fact that the space in which we operate is a metric space. Then we introduce topology and we can define limits using neighbourhoods, and we see that any metric space is a topological space with a countable base of neighbourhoods (i.e. is a first-countable space). For a topological space that is not first-countable we can use nets and filters that generalize the concept of sequence, but we are always in a topological space. There is some way to define the concept of limit of a sequence in a set where a topology is not defined?

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  • $\begingroup$ Please tell us which properties of a limit your definition should have. Because names are only names, in mathematics. $\endgroup$ – Siminore Feb 7 '15 at 11:42
  • $\begingroup$ There are limits and colimits in category theory. $\endgroup$ – Lehs Feb 7 '15 at 11:42
  • $\begingroup$ There are all sorts of weakenings of the notion of a topology, several of which were studied in the 1910s and 1920s when the most useful notion of what a topological space is was still in flux. Try googling "generalized topology" {AND} "limit". $\endgroup$ – Dave L. Renfro Feb 7 '15 at 11:57
  • $\begingroup$ @Siminore: The problem is just that! We have an intuitive notion of limits as ''some process that approximates something'', but this intuition is mathematically well-definable only in a topological space? $\endgroup$ – Emilio Novati Feb 7 '15 at 12:32
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    $\begingroup$ There are some limit concepts defined in terms of order rather than topology. For example: a sequence $A_n$ of sets converges to $A$ if $\bigcup_{k=1}^\infty \bigcap_{n=k}^\infty A_n =\bigcap_{k=1}^\infty \bigcup_{n=k}^\infty A_n = A$. $\endgroup$ – GEdgar Feb 7 '15 at 15:21
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Look up convergence space (e.g. here). This axiomatises the convergence of filters (or nets).

There is even a notion of convergence space just for sequences (instead of nets or filters). Fréchet defined these first, I think. Here the notion of convergence of a sequence is axiomatised. There are then special convergence spaces where the convergence is induced by a topology.

In a convergence space we have a few basic axioms:

  • the constant sequence converges to the constant.
  • if we modify a sequence by finitely many terms, and one sequence converges to some limit, the other one does too.
  • if a sequence converges to a limit, so does every subsequence of that sequence.

(I believe these are all the original ones.)

In order to be induced by a topology, we need the axiom that : if a sequence $(x_n)$ is such that every subsequence of it has itself a subsequence that converges to $x$, then the original sequence $(x_n)$ converges to $x$. Such a topology then need not be unique (as topologies need not be uniquely determined by the convergence of its sequences).

This nice answer assumes the last axiom (as L3) as well, and shows how to define a topology from it, and gives some references. There is another axiom to make it even more well-behaved (his axiom L4).

For nets we have similar axioms, and for filters too. And there we have exact characterisations for when the convergence is topological, as these convergence notions do chacterise topologies (again, one needs some extra axioms as for sequences). Kelly's book on Topology has some theory on this, IIRC. The sequential convergence spaces theory is somewhat obscure and mostly historically relevant (when they were not in agreement about what the right generalisation for a topology was).

Indeed, it is classical that convergence almost everywhere on Euclidean Lebesgue measure spaces has the property that it defines a convergence space that cannot be induced by a topology (as it does not satisfy the special subsequence of subsequence axiom).

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