  We consider Schmidt's game on the space of compact subsets of a given metric
space equipped with the Hausdorff metric, and the space of continuous functions
equipped with the supremum norm. We are interested in determining the generic
behaviour of objects in a metric space, mostly in the context of fractal
dimensions, and the notion of `generic' we adopt is that of being winning for
Schmidt's game. We find properties whose corresponding sets are winning for
Schmidt's game that are starkly different from previously established, and
well-known, properties which are generic in other contexts, such as being
residual or of full measure.
