On properties close to Polish cofinality in spaces of Baire-one and continuous functions

In this paper we prove that $B_1(X)$ has a dense cofinally Polish subspace if and only if $X$ is a ${\mathrm{\Delta }}_1$-space. Also we prove that $B_1(X,[0,1])$ has a dense cofinally Polish subspace if and only if $X$ is a weak $\lambda$-space. We also establish that $C_p(X,[0,1])$ is pseudocompact if and only if $C_p(X,[0,1])$ has a dense cofinally Polish subspace. This result provides answers to four open questions from (Tkachuk in RACSAM 115(68), 2021).