Can you define "limit" if you have only a topology or you need a metric?

Answer 1

You can define the limit of a sequence for any topology. You just say that a sequence x_n converges to x if every neighborhood of x contains a tail of the sequence. Unless the topology is Hausdorff (i.e. T_2), limits won't be unique.

Also, if you restrict yourself to sequences, don't expect many of the results you see in the metric space situation. For example, if x is in the closure of the set A, don't expect to have a sequence in A converging to x. Nor is compactness the same as every sequence having a convergent subsequence. To regain these results, limits of things called 'nets' are used. Essentially, a net allows an index set other than the natural numbers. That index set has to be a directed set so we can tell when something is 'further out' in the net. The point is that the natural numbers are only countable and that doesn't allow enough flexibility in general topological spaces to define limits as they need to be defined. Yes, the Hausdorff property is still needed for uniqueness of limits of nets.

In addition to nets, there are also things called filters that have limits. The limits of nets and the limits of filters are closely related. Willard's Topology book has a good treatment of these concepts.

Answer 2

It depends on the Topology. If you have a Hausdorf (or even T1) topology, then yes:

Define the limit of a sequence a1, a2, a3, . . . as a if for every n in N CHANGE (there exists a neighbourhood Un of a such that) TO (and EVERY neighbourhood U), ai is in Un for every i≥n.

So should read . . . define limit as a if for every n in N and every neighbourhood U of a, ai is in U for all i≥n.


If you have a set X={a,b,c} and a topology on X consisting of the sets {Ø, {a,b}, X} and the sequence a, a, a, a, a, a, . . . then you can't define the limit, because the limit could be a or b. That is why you need some sort of seperation structure (This set X is not T1, and thus not hausdorf).

Random may be correct about the limit not being required to be unique. I'm really not sure.

He is absolutely correct about Munkres' being a good book.

Answer 3

Yes.

First in an arbitrary topological space a sequence of points converges to x provided that for some point in the sequence, every point from that point on is in the neighborhood x. NOTE: this means that sequences can converge to more than one point in some topologies!

Next, a topological space is called a Hausdorff space if for each pair of distinct points there exists disjoint neighborhoods for the points.

So, if a topological space is endowed with Hausdorffness it is easy to see that a sequence can converge to at most one point. We call this point the limit of the sequence.

If I didn't make something clear just let me know.