A note on bounds of cardinal functions in the closed preimage

Abstract

Let X and Y be T1 spaces and f: X → Y be a closed and onto mapping. If a fiber of the mapping f is defined to be the inverse image of a singleton in the range, then a bound for the tightness of the domain is the product of the tightness of the range and the supremum of the tightness of the fibers of f. Similar bounds can also be shown for the Lindelöf degree and the extent of X. Examples are provided to demonstrate that such results are not possible for open maps. Cellularity and spread are discussed briefly.