I don't normally encounter smash products in the stuff I do, but rn I'm looking at some point-set properties of what are apparently smash products. My impression is that smash products usually arise in categorical constructions or when you're dealing with some generalized manifolds and want to get a nicer, related space (sorta like how suspensions are used) to study properties of.
Have smash products been studied very much when the spaces aren't as nice? A continuum is a compact, connected metric space. Two questions:
If $X$ and $Y$ are continua, is $X \wedge Y$ aposyndetic? Kelley?
If $X$ and $Y$ are continua, what is the hyperspace of $X \wedge Y$?
The "hyperspace" is the collection of closed subsets under the Hausdorff metric. Usually the hyperspace is very large, so a better question is probably this:
2') If $X$ and $Y$ are continua, what is $C(X \wedge Y)$?
Here $C(X \wedge Y)$ is the subset of the hyperspace consisting of the closed, connected subsets.
I'm wondering if we can decompose the space nicely assuming the smash is taken at cut points of $X, Y$ respectively, or if there are other constructions that allow extra-nice decomps. Any ideas or alternatives to get nice, comprehensible spaces?
I'm also curious to know about any strange and unintuitive (or instructive) examples of general, topological smash products when the spaces aren't super-nice.