A function $f:X\to Y$ between topological spaces is called $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\restriction}C$ is continuous.

A typical example of a function (of the first Baire class) which is not $\sigma$-continuous is the Pawlikowski function $P:(\omega+1)^\omega\to\omega^\omega$ (which is defined as the countable power $P=f^\omega$ of a bijection $f:\omega+1\to\omega$).

Let $\mathcal I_P$ be the $\sigma$-ideal of subsets $X$ of the compact metrizable space $(\omega+1)^\omega$ such that $P{\restriction}X$ is $\sigma$-continuous.

I am interesting in evaluating the standard cardinal characteristics $\mathrm{add}(\mathcal I_P)$, $\mathrm{cov}(\mathcal I_P)$, $\mathrm{non}(\mathcal I_P)$, $\mathrm{cof}(\mathcal I_P)$ of the $\sigma$-ideal $\mathcal I_P$.

It seems that among these four cardinal characteristics only the covering number $\mathrm{cov}(\mathcal I_P)$ was studied in the literature. In particular, Cichon, Morayne, Pawlikowski and Solecki proved that $\mathrm{cov}(\mathcal I_P)\ge\mathrm{cov}(\mathcal M)$ where $\mathcal M$ is the $\sigma$-ideal of meager subset in $\mathbb R$. Steprans gave a combinatorial description of the ideal $\mathcal I_P$ and proved the consistency of the strict inequality $\mathrm{cov}(\mathcal I_P)>\mathrm{cov}(\mathcal M)$. Steprans also observed that for every $A\in\mathcal I_P$ the image $P(A)$ is a meager subset of $\omega^\omega$, which implies that $\mathrm{cov}(\mathcal I_P)\ge\mathrm{cov}(\mathcal M)$ and $\mathrm{non}(\mathcal I_P)\le\mathrm{non}(\mathcal M)$. On the other hand, it can be shown that $\mathrm{non}(\mathcal I_P)\ge\mathfrak p$.

It is well-known that the strict inequality $\mathfrak p<\mathrm{non}(\mathcal M)$ is consistent. In particular, according to Table 4 in the survey paper of Blass, this strict inequality holds in the random, Hechler, Laver, and Mathias forcing models.

Problem 1. Which of the inequalities $\mathfrak p<\mathrm{non}(\mathcal I_P)$ and $\mathrm{non}(\mathcal I_P)<\mathrm{non}(\mathcal M)$ is consistent?

Problem 2. What is the value of $\mathrm{non}(\mathcal I_P)$ (and other cardinal characteristics of the ideal $\mathcal I_P)$ in the random, Hechler, Laver, and Mathias forcing models?


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