Cauchy Sequences in General Topological Spaces

[ebola1717]

"Cauchy" Sequences in General Topological Spaces

Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every neighborhood around the limit, the sequence is eventually entirely contained in that neighborhood. But this requires the sequence converges to a point in the space. For incomplete spaces, (does this idea exist in general topological spaces?) not all sequences that get close converge in the space.

For Cauchy sequence, then, it sounds like we want that for sufficiently large N and n>N, if m>n, then a_m is contained in a neighborhood around a_n, and that these neighborhoods get smaller, in some sense. My thought process lead me to believe we might be able to define "smaller" by strict inclusion, but it seems like the neighborhoods can stay pretty large in this case. In any case, is there a way we can define an analogous notion to Cauchy sequences in general topological spaces?

The motivation for this was that I was thinking about the definitions of compactness, specifically whether sequential compactness was equivalent to limit point compactness and normal compactness in topological spaces with weaker separation axioms than metrizability. I've been lead to believe the answer is no, and this seemed like a logical place to start.

 

[Landau]

The familiar definition of Cauchy sequence talks about the difference of two sequence elements. In a general topological space the notion of difference of course does not exist. So you need extra structure. Of course it works in a topological group:

A net (x_i)_i such that for every neighbourhood V of the identity there exists N such that x_ix_j\^{-1}\in V whenever i,j \geq N.

But we can do with a bit less: we only need a uniform structure. For a uniform space, we can define a Cauchy net (recall that in general, a natural generalization of sequences - which are countable things - are nets):

A net (x_i)_i such that for every entourage (=element of the uniform structure) V there exists N such that (x_i,x_j)\in V whenever i,j \geq N.

(Indeed, a topological group G is a uniform space if we let the entourages be those subsets V of GxG for which there exists a neighbourhood U of the identity such that \{(x,y)\ :\ xy^{-1}\in U\}\subseteq V.)

 

[Greg Bernhardt]

Yes, there is an equivalent of a Cauchy sequence in general topological spaces. It is called a "Cauchy net" or "Cauchy filter." These are generalizations of Cauchy sequences that do not necessarily converge to a point in the space. Instead, they converge to a "limit" point which may not be in the space itself.

To define a Cauchy net, we start with a directed set (often denoted as I) which is a set with a partial ordering relation that is reflexive, transitive, and satisfies the property that for any two elements, there exists a third element that is greater than both. Then, we define a net as a function from I to the space X. The net is said to be Cauchy if for any neighborhood U of the limit point, there exists an element in I such that all elements after that element in the directed set are mapped into U.

Similarly, a Cauchy filter is a filter on a set I such that for any neighborhood U of the limit point, there exists an element in the filter such that all elements after that element are contained in U.

These definitions may seem a bit abstract, but they capture the idea of a sequence getting "closer and closer" to a limit point without actually converging to it. And yes, these notions do exist in general topological spaces, not just in complete spaces.

As for your motivation about compactness, it turns out that sequential compactness is equivalent to limit point compactness in any topological space, regardless of the separation axioms. However, normal compactness is not equivalent to sequential or limit point compactness in general.

I hope this helps clarify the concept of Cauchy sequences in general topological spaces!