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Would it have sense to defined Cauchy sequence in non metric space ? (for example, if $(X,T)$ is a topology space : $$\forall U\in T, 0\in U, \exists N\in \mathbb N: x_n-x_m\in U.$$

And if yes, would it be interesting ?

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    $\begingroup$ What do you mean by $x-y$ in a general topological space? Does it make sense? $\endgroup$ – mfl Nov 13 '16 at 15:27
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As you wrote it, no, this does not make sense, because there is no subtraction operation $x_n - x_m$ in a metric space.

However, there is a theory of uniform spaces, which is a special kind of topological space $X$ which need not be metrizable but which nonetheless has a built in theory of Cauchy nets. One has to use nets instead of sequences, because unlike metric spaces where sequences are sufficient to detect limit points of a subset, one must use nets to detect limit points in general nonmetrizable spaces.

The rough idea of a uniform space is that in place of the collection of inequalities "$d(x,y) < r$" for $r > 0$, one introduces collection of symmetric, reflexive relations $U \subset X \times X$ called entourages. In place of the condition that $d(x,y)=0 \implies x=y$ one has the condition that the intersection of all entourages is the diagonal subspace $\Delta = \{(x,x) \in X \times X \bigm| x \in X\}$. More axioms are needed in order to say how to express open sets of the topology in terms of the collection of entourages, to replace how open sets of a metric topology are expressed in terms of open balls. For example, some axiom is needed which will replace the triangle inequality and still allow you to construct a basis for a topology, as the triangle inequality lets you do in metric spaces.

See the great old book "General Topology" by Kelley for a full treatment of uniform spaces, or see the wikipedia page for a bare bones outline.

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  • $\begingroup$ For those who, like me, don’t care much for Kelley, Willard’s General Topology also has a good treatment of uniform spaces. $\endgroup$ – Brian M. Scott Nov 13 '16 at 18:08
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If (xα) is a sequence from $\mathbb N$ into X, and if Y is a subset of X, then we say that (xα) is eventually in Y (or residually in Y) if there exists an α in A so that for every β in A with β ≥ α, the point xβ lies in Y.

If (xα) is a sequence in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x if and only if for every neighbourhood U of x, (xα) is eventually in U. So you can define a notion of convergence also in a not metric space. We can't extend this notion to something similar to a Cauchy sequence, because it doesn't make sense to write something as $x-y$ if we haven't a metric notion.

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