I don't actually know much about topology, apart from the background level, and the edge-case of graph theory.
Often the coarse topology of networks is of interest, or the more metaphorical knowledge topology, or actual continuous fields or knots or whatever.
Sometimes I care about the induced topology of a metric convergence, but this is hardly fancy stuff.
Apparently grown-up topology has uses too though?
Ghrist, R. W. (2014). Elementary applied topology. Online.