Background
A CW complex is a Hausdorff space and it is the union of its some of its subsets called cells, and cells are homeomorphic images in $X$ of some closed $k$-balls.
The weak topology of a CW complex X is defined as the topology having the property that a subset of $X$ is closed if and only if it is closed in each cell of $X$.
The question
What is the motivation of requiring that the topology is weak? What is, if $X$ has more closed sets then in this definition, and what is if it has less. And why are closed sets are generally used in this definition, why not open sets? (I saw in some places this definition with open sets. Is it an error, or it is an equivalent definition?)
Best Answer
This is an answer to the motivation for the definition of CW-complex. The main one is given by the following from Section $5$ of Whitehead's paper Combinatorial Homotopy I.
Let $K$ be a CW-complex.
(A) A map $f: X \to Y$ of a closed (open) subset, $ X \subset K$ in any space, $Y$, is continuous provided $f| X \cap \bar{e}$ is continuous for each cell $e \in K$.
Nowadays we might do it a bit differently and say that the topology on $K$ is defined so that this property holds. This can be done by defining the $n$ skeleton $K^n$ to be obtained from the $(n-1)$-skeleton $K^{n-1}$ by attaching $n$-cells $e^n_\lambda$ by maps $a_\lambda: S^{n-1} \to K^{n-1}$. Then define the topology on $K$ so that a map $f: K \to X$ is continuous if and only if the restriction of $f$ to $K^n \to Y$ is continuous for all $n \geqslant 0$.
This accords with the modern view that topology is "about" the category of spaces and continuous maps, and to construct these maps we often use the universal properties of pushouts and colimits (and for other examples, pullbacks and limits).
A gentle introduction to adjunction spaces and finite cell complexes is given in the book Topology and Groupoids.
Ioan James once said at Oxford that Whitehead took a year to prove the product property (H), which is now subsumed under the notion of convenient category of spaces.
His motivation can also be seen in his highly original 1941 papers in the Annals of Math, referred to in CHI. The latter paper, and CHII and Simple Homotopy Types, constitute a rewrite and extension of these essentially prewar papers.