I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, plus some other requirements. They are endowed with the weak topology. That is, a subset $A$ is closed in $X$ iff $A\cap X^n$ is closed for each $X^n$.
In functional analysis, I learned that the weak topology on a Banach space $Y$ is the coarsest topology making the elements in the dual space $Y^*$ continuous.
Do these two topologies share anything more than a name? Why is it they are both given the label 'weak'?