On spread and condensations

A space $X$ has a property ${\mathcal {P}}$ strictly if every finite power of $X$ has ${\mathcal {P}}$. A condensation is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of $X$ is countable, then $X$ can be condensed onto a strictly hereditarily separable space. If $s(C_{p}(X))\leq \omega$, then $C_{p}(X)$ can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of $C_{p}(X)$ is strictly hereditarily separable. Under $(MA+\neg CH)$, if $G$ is a topological group such that $s(C_{p}(G))\leq \omega$, then $G$ is strictly hereditarily Lindelöf and strictly hereditarily separable.