Does there exists two cardinal functions in topology, each of them not hereditary but their product is hereditary?
For cardinal function i mean a rule to associate a cardinal number to every topological space such that for homeomorphic spaces the result is the same, for example the weight (the minimal cardinality of a base), the density (the minimal cardinality of a dense subset), the Lindelöf degree (the minimal infinite cardinal $\kappa$ such that every open cover has a subcover of cardinality $\leq\kappa$) etc...
A cardinal function $\varphi$ is called hereditary if whenever $Y\subseteq X$, $\varphi(Y)\leq\varphi(X)$ holds. Examples of hereditary cardinal functions are the weight, the cardinality, examples of non-hereditary ones are the density, the Lindelöf degree, etc...