Convergence of sequences in topological spaces.

Question

To reference in my thesis, at first, I'd like a book of general topology that addressed convergence of sequences in topological spaces not necessarily metrizables​​. The concept seems plausible in Hausdorff topological spaces. See these notes for more.

The references I could get (as the books of John L. Kelley, MG Murdeshwar and Bourbaki) do not address sequences in topological spaces. In fact, Kelley's book is a brief definition of sequences in first countable topological spaces. But its definition depends completely on the definition of convergence in nets. And I do not want to deal with convergence in nets.

• Question: Is there a book of general topology well accepted by the mathematical community to define convergence in topological spaces without speaking nets?

• Question: Is there any research article that talks about convergence of sequences in topological spaces?

Thank's.

Answer 1

If you look at Munkres' Topology textbook (2000 edition p.98), a definition of a convergent sequence in an arbitrary topological space is given as follows. A sequence $x_1, x_2, \ldots$ of points in a space $X$ converges to a point $x \in X$ if for each neighborhood $U$ of $x$, $\exists N$ such that $\forall n \geqslant N$, $x_n \in U$.

The topology here is arbitrary and there is no mention of nets.

Answer 2

It is no problem to define convergence of sequences in any topological space: Learner's answer above does just this, as do many standard texts.

However, I want to make the point that convergence of sequences is not really the correct notion in an arbitrary topological space. For instance, the closure of a subset in an arbitrary topological space need not be the set of limits of convergent sequences with values in that subset. Thus in general the notion of sequential closure differs from that of closure. In fact there are topological spaces which are not discrete but sequentially discrete: the only convergent sequences are the eventually constant ones (in fact, certain non-Archimedean ordered fields have this property).

This is why most standard texts in general topology move on swiftly to discuss nets (and/or filters). I have a set of notes on convergence available here, and indeed it is mostly a discussion of nets, then filters, then the relationship between nets and filters. But I begin with a discussion of sequences, including a treatment of why sequences are not adequate for discussing convergence in general topological spaces.

Let me also say that it is nevertheless the case that the (surprisingly complicated) behavior of sequences in arbitrary topological spaces has been studied. A landmark paper in this area is S.P. Franklin's 1965 Spaces in which sequences suffice, which led to the notion of a sequential space: this is precisely the condition in which sequentially closed subsets are closed.

Answer 3

I'll mention my own paper

R. Brown, ``Sequentially proper maps and a sequential compactification'', London Math Soc. (2) 7 (1973) 515-522.

and also

Johnstone, P. T. On a topological topos. Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.

Booth, P.; Tillotson, J. Monoidal closed, Cartesian closed and convenient categories of topological spaces. Pacific J. Math. 88 (1980), no. 1, 35–53.

These all discuss the use of sequences in topology, but I expect there are lots more.

I thought of the idea for my paper after teaching a course in analysis and realising how nice were the sequence proofs of some basic facts, such as properties of closed, bounded subsets of Euclidean space. Why go to greater generality than is needed in the context in which one is working, or teaching? For basic analysis, subsets of normed spaces, with their metric, are often all one needs.