In this previous question it is asked how to prove, given a sequence $x_1,x_2, \ldots$ of points in a product space $\prod X_\alpha$, that the sequence converges to the point $x$ if and only if the sequence $\pi_\alpha (x_1), \pi_\alpha (x_2)\ldots$ converges to $\pi_\alpha (x)$ for each $\alpha$.
Question:
Are we actually allowed to talk about a sequence of points in the product topology?
The point is that I think we can, if we assume that the topological spaces are – for example – metrizable. However, in the general case we should rather talk about nets or filters. Am I right?
Notice that this question arises because in Munkres it is proved by means of sequences, but in Willard by means of nets, which I think should be conceptually more appropriate.
Any feedback is most appreciated.
Thank you for your time.
A sequence $\{x_n\}$ converges to a point $x$ if, for each open set $U$ containing $x,$ we have $x_n\in U$ for all sufficiently large $n.$ This definition is topological; it makes sense in any topological space.
It is true that sequential convergence does not have the same significance in general topological spaces as it does in metric spaces. For example, in a topological space, you may not be able to get the closure of a set by taking all limits of convergent sequences in the sets. But this does not mean that sequential convergence is of no use at all in general topology, and it certainly does not mean that you aren't allowed to talk about sequences. (Especially if you happen to live in the United States, where he have something called the First Amendment.)
In your question, I don't know what is the "it" which Munkres proves by means of sequences and Willard proves by means of nets. A proof is a proof, and if you can prove something just using sequences, that seems preferable to a proof of the same thing using nets. But maybe you meant to say that Willard proves some statement about nets and Munkres only proved it for the special case of sequences?