Cellularity of product space Let $\{X_s\}_{s\in S}$ be a family of topological space, and $d(X_s)\leq m$, then the cellularity, i.e., the supremum of the cardinalities of all families of pairwise disjoint non-empty open subsets of the product space $\prod_{s \in S} X_s$, is smaller or equal than $m$. (End of the theorem.)
I have referred two books, one is theorem 2.3.17 of Engelking's General Topology. They all have one set saying that $|T|\leq 2^m$. In Engelking's, he let $\{U_t\}_{t\in T}$ is a collection of disjoint nonempty open subsets. Then he calim that $|T|\leq 2^m$. But I did not see how this works.
 A: So we have a family $U_t, t \in T$ of pairwise disjoint (basic) open subsets of $\prod_{S \in S} X_s$, where all $X_s$ obey $d(X_s) \le m$.
We want to show that $|T| \le m$, so Engelking assumes that $|T| > m$. But a subfamily of a pairwise disjoint family is still pairwise disjoint so we can assume  that $|T| \le 2^m$ because we know that $m < 2^m$ and if we'd have more we throw the rest away and the argument to get a contradiction can go on. 
I.e. to put it in small numbers: if you know you have a bad set of size 6 (you want to show a bad set has size at most 5) you might as well assume it has size at most 10, if that's convenient and if throwing members away cannot make the bad property (pairwise disjointness here) go away. 
Engelking needs it for applying the HMP-theorem from just before, so that's where the bound $2^m$ is convenient for him. 
$\kappa > m$ implies $\kappa \ge m^+$ and $m^+ \le 2^m$, for infinite cardinals. 
There is an alternative proof (Kunen's book set theory has that one, as has Juhasz books on cardinal functions), using the $\Delta$-system lemma.
See this blog post for an exposition of that idea.
It gives a more general result: in the above product, if every finite subproduct (over any finite index subset $F$) has $c(\prod_{s \in F} X_s)\le m$ then $c(\prod_{s in S} X_s) \le m$ too. The finite subproducts determine the result (and basic open sets are really open sets in finite subproducts) (note that cellularity need need not be preserved by even products of two spaces), while $d(X_s) \le m$ does imply $c(\prod_{s \in F}) \le d(\prod_{s \in S} X_s) \le m$ for all finite subproducts and so we can apply it in this common case. It's a very useful (and interesting) fact that a product like $\mathbb{R}^\kappa$ is ccc regardless of $\kappa$ while separability is lost for $\kappa>   2^{\aleph_0}$.
