# Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function

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?