TITLE: Convergence in product topology
QUESTION [18 upvotes]: Let $x_1,x_2, \ldots$ be a sequence of points of the product space $\prod X_\alpha$. Show 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 $\alpha$. Is this fact true for box topology instead of product topology?
But what does it mean to converge in product topology? Is it that a common neighborhood exists for all points in the sequence?

REPLY [5 votes]: Here is my usual abstract-nonsense answer.

The product of topological spaces satisfies (or rather, is defined by) the universal property: A continuous map into the product is nothing else than a family of continuous maps into the factors.
A sequence in a space $X$ is a (continuous) map $\mathbb{N} \to X$, and it converges iff this map has an extension to a continuous map $\mathbb{N}^+ \to X$, where $\mathbb{N}^+ = \mathbb{N} \cup \{\infty\}$ is the one-point compactification, where the basic-open neighborhoods of $\infty$ are cofinite. The value at $\infty$ is the limit of the sequence.
From 1. we get $\hom(\mathbb{N}^+,\prod_\alpha X_\alpha) = \prod_\alpha \hom(\mathbb{N}^+,X_\alpha)$, and from 2. we see that this means that a sequence in $\prod_\alpha X_\alpha$ is just an $\alpha$-indexed family of sequences in the $X_\alpha$, and that a limit of this sequence is the same as the $\alpha$-indexed family of some limits of the individual sequences.