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'?
The topology on a CW complex $X$ is the final topology on $X$ with respect to the inclusions $X^n \to X$, ie. it's the finest topology on $X$ such that these inclusions are continuous.
On the other hand, as you say the weak topology is the initial topology wrt continuous linear functionals. These two notions are dual (initial and final topologies).
I doubt there's more of a connection than that. These are two different fields of mathematics (algebraic topology vs "analysis").