$\sigma$-Continuous functions and related cardinal characteristics of the continuum

TL;DR

This paper investigates $\sigma$-continuous functions and introduces cardinal characteristics of the continuum, establishing relationships and equalities among them, which resolve previous open problems in the field.

Contribution

The paper defines new cardinal invariants related to $\sigma$-continuity and proves their exact relationships, resolving an open problem about these characteristics.

Findings

Established that $rak p \, extless\, rak q_0=\mathfrak c_{\bar\sigma}$

Proved that $rak c_{\bar\sigma} = \min\{rak c_{\sigma}, \frak b, \frak q\}$

Showed inequalities linking these invariants to classical cardinal characteristics

Abstract

A function $f:X\to Y$ between topological spaces is called $\sigma$-$continuous$ (resp. $\bar\sigma$-$continuous$) if there exists a (closed) cover $\{X_n\}_{n\in\omega}$ of $X$ such that for every $n\in\omega$ the restriction $f{\restriction}X_n$ is continuous. By $\mathfrak c_\sigma$ (resp. $\mathfrak c_{\bar\sigma}$) we denote the largest cardinal $\kappa\le\mathfrak c$ such that every function $f:X\to\mathbb R$ defined on a subset $X\subset\mathbb R$ of cardinality $|X|<\kappa$ is $\sigma$-continuous (resp. $\bar\sigma$-continuous). It is clear that $\omega_1\le\mathfrak c_{\bar\sigma}\le\mathfrak c_\sigma\le\mathfrak c$. We prove that $\mathfrak p\le\mathfrak q_0=\mathfrak c_{\bar\sigma}=\min\{\mathfrak c_\sigma,\mathfrak b,\mathfrak q\}\le\mathfrak c_\sigma\le\min\{\mathrm{non}(\mathcal M),\mathrm{non}(\mathcal N)\}$. The equality $\mathfrak c_{\bar\sigma}=\mathfrak q_0$ resolves a problem from the initial version of the paper.

Equations

$$\mathcal{I}_{P}=\{X\subset(\omega+1)^{\omega}:\mbox{$P{\restriction}X$ is not $\sigma$-continuous}\}$$

$${\omega }_1\le {\mathfrak{c}}_{\bar{\sigma }}\le {\mathfrak{c}}_{\sigma }\le \mathfrak{c},$$

$$\mathfrak{p}\le {\mathfrak{q}}_0\le \min \{\mathfrak{b},\mathrm{non}(\mathcal{N}),\mathfrak{q}\}\le \mathfrak{q}\le \log ({\mathfrak{c}}^{+}),$$

$${\omega }_1\le {\mathfrak{q}}_2\le {\mathfrak{q}}_1\le {\mathfrak{q}}_0\le \mathfrak{q}\le \log ({\mathfrak{c}}^{+})\le \mathfrak{c}.$$

$${\mathcal{F}}_{A,B,n}=\bigcap _{a\in A}\{F\in [2^{<\omega }{]}^{<\omega }:F\cap \{a\upharpoonright i{\}}_{i\in \omega }=\varnothing \}\cap \bigcap _{b\in B}\{F\in [2^{<\omega }{]}^{<\omega }:|F\cap \{b\upharpoonright i{\}}_{i\in \omega }|\ge n\}.$$

$$\mathcal{F}=\{{\mathcal{F}}_{A,B,n}:A\in \mathcal{A},B\in \mathcal{B},n\in \omega \}$$

$$K_n=\bigcap _{k\ge n}\{x\in 2^{\omega }:F_k\cap \{x\upharpoonright i{\}}_{i\in \omega }=\varnothing \}=\bigcap _{k\ge n}\bigcap _{i\in \omega }\{x\in 2^{\omega }:x\upharpoonright i\notin F_k\}$$

$$f_C(C\cap A)=f(C\cap A)\subset {\Sigma }_C\cap Y\subset f_C(\tilde{C})\cap K_C\cap Y\subset K_C\cap Y=f(C\cap A)$$

$$\Lambda \cap X=\bigcup _{C\in \mathcal{C}}{\Lambda }_C\cap X=\bigcup _{C\in \mathcal{C}}C\cap A=A,$$

$$0={\mu }_C(A)=\lambda (g_C^{-1}(A)\setminus P)=\lambda (B\cap \tilde{C}\setminus P).$$

$$\lambda (B)=\lambda (B\cap P)+\lambda (B\setminus P)=0+0=0$$

$${\mathfrak{c}}_{\bar{\sigma }}<{\mathfrak{c}}_{\sigma },\mathfrak{p}<{\mathfrak{q}}_2,{\mathfrak{q}}_2<{\mathfrak{q}}_1,{\mathfrak{q}}_1<{\mathfrak{q}}_0$$

$${\mathfrak{b}}_{{\omega }_1}=\mathfrak{b},{\mathfrak{b}}_{{\mathfrak{c}}^{+}}={\omega }_1,{\mathfrak{b}}_{\mathfrak{p}}\ge \mathfrak{p},{\mathfrak{b}}_{{\mathfrak{q}}_1}\ge {\mathfrak{q}}_1,\text{mbox}and{\mathfrak{q}}_1=\max \{\kappa :\kappa \le {\mathfrak{b}}_{\kappa }\}.$$

Full text

In this paper we introduce and study two cardinal characteristics of the continuum, related to $\sigma$-continuity.

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

The problem of finding a (Borel) function $f:\mathbb{R}\to\mathbb{R}$ which is not $\sigma$-continuous is not trivial and was first asked by Lusin. This problem of Lusin was answered by Sierpiński [16], [17] (under CH), Keldyš [12], and later by Adyan, Novikov [1], van Mill and Pol [13], Jackson, Mauldin [10], Cichoń, Morayne, Pawlikowski, and Solecki [4], [18], Darji [5]. By a dichotomy of Zapletal [21] (generalized by Pawlikowski and Sabok [15]), a Borel function $f:X\to Y$ between metrizable analytic spaces is not $\sigma$-continuous if and only if $f$ contains a topological copy of the Pawlikowski function $P:(\omega+1)^{\omega}\to\omega^{\omega}$ (which is the countable power of a bijection $\omega+1\to\omega$). Since the Pawlikowski function $P$ is not $\sigma$-continuous, the family

[TABLE]

is a proper $\sigma$-ideal of subsets of the compact metrizable space $(\omega+1)^{\omega}$.

Let $\mathfrak{c}_{\sigma}$ (resp. $\mathfrak{c}_{\bar{\sigma}}$) be the largest cardinal $\kappa$ such that every function $f:X\to\mathbb{R}$ defined on a subset $X\subset\mathbb{R}$ of cardinality $|X|<\kappa$ is $\sigma$-continuous (resp. $\bar{\sigma}$-continuous). It is clear that

[TABLE]

so $\mathfrak{c}_{\bar{\sigma}}$ and $\mathfrak{c}_{\sigma}$ are typical small uncountable cardinals in the interval $[\omega_{1},\mathfrak{c}]$.

In this paper we establish the relation of the cardinals $\mathfrak{c}_{\bar{\sigma}}$ and $\mathfrak{c}_{\sigma}$ to some known cardinal characteristics of the continuum: $\mathrm{non}(\mathcal{M})$, $\mathrm{non}(\mathcal{N})$, $\mathfrak{p}$, $\mathfrak{b}$, $\mathfrak{q}_{0}$, $\mathfrak{q}$.

Let us define their definitions. By $\omega$ we denote the smallest infinite cardinal, by $2^{\omega}$ the Cantor cube $\{0,1\}^{\omega}$, and by $\mathfrak{c}$ the cardinality of $2^{\omega}$. Let $\mathcal{M}$ be the ideal of meager sets in the real line, $\mathcal{N}$ be the ideal of Lebesgue null sets in $\mathbb{R}$, and $\mathcal{K}_{\sigma}$ be the $\sigma$-ideal, generated by compact subsets of $\omega^{\omega}$. For a family of sets $\mathcal{I}$ with $\bigcup\mathcal{I}\notin\mathcal{I}$ let $\mathrm{non}(\mathcal{I})=\min\{|A|:A\subset\bigcup\mathcal{I}\;\;A\notin\mathcal{I}\}$. The cardinal $\mathrm{non}(\mathcal{K}_{\sigma})$ is usually denoted by $\mathfrak{b}$, see [3].

Let $\mathfrak{p}$ be the smallest cardinality of a family $\mathcal{F}$ of infinite subsets of $\omega$ such that every finite subfamily $\mathcal{E}\subset\mathcal{F}$ has infinite intersection and for any infinite set $I\subset\omega$ there exists a set $F\in\mathcal{F}$ such that $I\setminus F$ is infinite.

It is known [6], [20], [3] that $\mathfrak{p}\leq\mathfrak{b}\leq\mathrm{non}(\mathcal{M})\leq\mathfrak{c}$, and $\mathfrak{p}=\mathfrak{c}$ is equivalent to Martin’s Axiom for $\sigma$-centered posets.

A subset $A\subset\mathbb{R}$ is called a $Q$-set if each subset of $A$ is of type $F_{\sigma}$ in $A$. Let

• •

$\mathfrak{q}_{0}$ be the smallest cardinality of a subset $A\subset\mathbb{R}$, which is not a $Q$-set, and

• •

$\mathfrak{q}$ be the smallest cardinal $\kappa$ such that every subset $X\subset\mathbb{R}$ of cardinality $|X|\geq\kappa$ is not a $Q$-set.

It is known [2] that

[TABLE]

where $\log\mathfrak{c}^{+}=\min\{\kappa:2^{\kappa}>\mathfrak{c}\}$. More information on $Q$-sets and the cardinals $\mathfrak{q}_{0}$ and $\mathfrak{q}$ can be found in [14, §4] and [2]. The following theorem was proved on 12.09.2019 during the XXXIII International Summer Conference on Real Function Theory in Ustka at the Baltic sea.

By [11, 13.5], there exists a continuous bijective map $g:\tilde{Z}\to\tilde{X}$ from a Polish space $\tilde{Z}$ such that for every $n\in\omega$ the Borel set $g^{-1}(B_{n})$ is open in $\tilde{Z}$. Consider the subspace $Z=g^{-1}(X)$ of the Polish space $\tilde{Z}$ and observe that the map $h=f\circ g{\restriction}Z$ is continuous. Indeed, for any $n\in\omega$ the preimage $h^{-1}(V_{n})$ of the basic open set $V_{n}$ equals $Z\cap g^{-1}(B_{n})$ and hence is open in $Z$.

Since $|Z|=|g^{-1}(X)|=|X|<\mathfrak{b}$, the set $Z=g^{-1}(X)$ is contained in some $\sigma$-compact subset $A$ of $\tilde{Z}$. Write the set $A$ as the countable union $A=\bigcup_{n\in\omega}A_{n}$ of compact sets $A_{n}$ in $\tilde{Z}$. For every $n\in\omega$ the image $K_{n}:=g(A_{n})\subset\tilde{X}$ of $A_{n}$ is compact and the continuous bijective function $g{\restriction}A_{n}:A_{n}\to K_{n}$ is a homeomorphism and so is the function $g{\restriction}A_{n}\cap Z:A_{n}\cap Z\to K_{n}\cap X$. Then the restriction $f{\restriction}K_{n}\cap X=f\circ g\circ(g{\restriction}A_{n}\cap Z)^{-1}$ is continuous, and the function $f$ is $\bar{\sigma}$-continuous. ∎

We say that two subsets $A,B$ of a topological space $X$ can be separated by $\sigma$-compact sets if there are two disjoint $\sigma$-compact subsets $\tilde{A}$ and $\tilde{B}$ of $X$ such that $A\subset\tilde{A}$ and $B\subset\tilde{B}$.

Having in mind the characterization of $\mathfrak{q}_{0}$ in Lemma 1, let us consider two modifications of $\mathfrak{q}_{0}$. Namely, let

• •

$\mathfrak{q}_{1}$ be the smallest cardinal $\kappa$ for which there exists a subset $A\subset 2^{\omega}$ of cardinality $|A|\leq\kappa$ and a family $\mathcal{B}$ of compact subsets of $2^{\omega}$ with $|\mathcal{B}|\leq\kappa$ such that the sets $A$ and $\bigcup\mathcal{B}$ are disjoint but cannot be separated by $\sigma$-compact sets in $2^{\omega}$;

• •

$\mathfrak{q}_{2}$ be the smallest cardinal $\kappa$ for which there exist families $\mathcal{A},\mathcal{B}$ of compact subsets of $2^{\omega}$ with $\max\{|\mathcal{A}|,|\mathcal{B}|\}\leq\kappa$ such that the sets $\bigcup\mathcal{A}$ and $\bigcup\mathcal{B}$ are disjoint but cannot be separated by $\sigma$-compact sets in $2^{\omega}$.

This definition and Lemma 1 imply that

[TABLE]

The following theorem is the main result of this paper.

The proof of this theorem is divided into a series of lemmas.

Let $[2^{<\omega}]^{<\omega}$ be the family of finite subsets of $2^{<\omega}$. For any disjoint compact subset $A,B\subset 2^{\omega}$ and $n\in\omega$ consider the set

[TABLE]

We claim that this set is infinite. Indeed, since $A,B$ are compact disjoint sets in $2^{\omega}$, there exists $m\in\omega$ such that $\mathrm{pr}_{m}(A)\cap\mathrm{pr}_{m}(B)=\emptyset$. Then for every $k\geq n$ the set $F_{k}=\bigcup_{i=m}^{m+k}\mathrm{pr}_{i}(B)$ belongs to the family $\mathcal{F}_{A,B,n}$.

We claim that the family

[TABLE]

is infinitely centered in the sense that each finite non-empty subfamily $\mathcal{E}\subset\mathcal{F}$ has infinite intersection. Indeed, write the family $\mathcal{E}$ as $\mathcal{E}=\{\mathcal{F}_{A_{i},B_{i},n_{i}}\}_{i=1}^{k}$ for some $A_{1},\dots,A_{k}\in\mathcal{A}$, $B_{1},\dots,B_{k}\in\mathcal{B}$, $n_{1},\dots,n_{k}\in\omega$. Consider the compact sets $A=\bigcup_{i=1}^{k}A_{i}$ and $B=\bigcup_{i=1}^{k}B_{i}$ and observe that they are disjoint as $A\cap B\subset(\bigcup\mathcal{A})\cap(\bigcup\mathcal{B})=\emptyset$. Let $n=\max_{1\leq i\leq k}n_{i}$ observe that $\mathcal{F}_{A,B,n}\subset\bigcap\mathcal{E}$, which implies that the intersection $\bigcap\mathcal{E}$ is infinite.

Since $|\mathcal{F}|=|\omega\times\mathcal{A}\times\mathcal{B}|<\mathfrak{p}$, the family $\mathcal{F}$ has infinite pseudointersection $\{F_{k}\}_{k\in\omega}\subset[2^{<\omega}]^{<\omega}$, which means that for every $A\in\mathcal{A}$, $B\in\mathcal{B}$ and $n\in\omega$ the set $\{k\in\omega:F_{k}\notin\mathcal{F}_{A,B,n}\}$ is finite.

For every $n\in\omega$ consider the closed subset

[TABLE]

of $2^{\omega}$ and observe that $\bigcup\mathcal{A}\subset\bigcup_{n\in\omega}K_{n}\subset 2^{\omega}\setminus\bigcup\mathcal{B}$. It follows that $2^{\omega}\setminus\bigcup_{n\in\omega}K_{n}$ is a Polish space containing the set $\bigcup\mathcal{B}$. The definition of the cardinal $\mathfrak{b}\geq\mathfrak{p}>|\mathcal{B}|$ ensures that $\bigcup\mathcal{B}$ is contained in some $\sigma$-compact subset $\Sigma$ of the Polish space $2^{\omega}\setminus\bigcup_{n\in\omega}K_{n}$. Then $\bigcup_{n\in\omega}K_{n}$ and $\Sigma$ are two disjoint $\sigma$-compact sets separating the sets $\bigcup\mathcal{A}$ and $\bigcup\mathcal{B}$ in the Cantor cube $2^{\omega}$. ∎

Assuming that $\mathfrak{c}_{\bar{\sigma}}<\mathfrak{q}_{0}$, we can find a subset $X\subset\mathbb{R}$ of cardinality $|X|=\mathfrak{c}_{\bar{\sigma}}<\mathfrak{q}_{0}$ and a function $g:X\to\mathbb{R}$ which is not $\bar{\sigma}$-continuous. The strict inequality $|X|<\mathfrak{q}_{0}$ implies that $X$ is a $Q$-space. Consequently, each subset of $X$ is of type $F_{\sigma}$ in $X$ and the function $g$ is Borel. Since $|X|<\mathfrak{q}_{0}\leq\mathfrak{b}$, we can apply Theorem 1 and conclude that the function $g$ is $\bar{\sigma}$-continuous, which contradicts the choice of $g$. This contradiction completes the proof of the equality $\mathfrak{c}_{\bar{\sigma}}=\mathfrak{q}_{0}$.

Since $\mathfrak{q}_{0}=\mathfrak{c}_{\bar{\sigma}}\leq\mathfrak{c}_{\sigma}$, the equality $\mathfrak{q}_{0}=\min\{\mathfrak{c}_{\sigma},\mathfrak{b},\mathfrak{q}\}$ will follow as soon as we prove that the strict inequality $\mathfrak{q}_{0}<\min\{\mathfrak{b},\mathfrak{q}\}$ implies $\mathfrak{c}_{\sigma}\leq\mathfrak{q}_{0}$. Assuming that $\mathfrak{q}_{0}<\min\{\mathfrak{q},\mathfrak{b}\}$, find a $Q$-set $Y\subset\mathbb{R}$ of cardinality $|Y|=\mathfrak{q}_{0}$. By the definition of $\mathfrak{q}_{0}$ there exists a subset $X\subset\mathbb{R}$ of cardinality $|X|=\mathfrak{q}_{0}$ which is not a $Q$-set. Let $f:X\to Y$ be any bijection. We claim that either $f$ or $f^{-1}$ is not $\sigma$-continuous. To derive a contradiction, assume that both maps $f$ and $f^{-1}$ are $\sigma$-continuous. Then there exists a countable cover $\mathcal{C}$ of $X$ such that for every $C\in\mathcal{C}$ the restriction $f{\restriction}C:C\to f(C)$ is a homeomorphism. By the Lavrentiev Theorem [11, 3.9], for every $C\in\mathcal{C}$ the topological embedding $f{\restriction}C$ can be extended to a topological embedding $f_{C}:\tilde{C}\to\mathbb{R}$ of some $G_{\delta}$-subset $\tilde{C}$ of $\mathbb{R}$. Since $X$ is not a $Q$-set, there exists a subset $A\subset X$ which is not of type $F_{\sigma}$ in $X$. On the other hand, for every $C\in\mathcal{C}$ the subset $f(C\cap A)$ is of type $F_{\sigma}$ in the $Q$-space $Y$. Then there exits a $\sigma$-compact set $K_{C}\subset\mathbb{R}$ such that $f(C\cap A)=Y\cap K_{C}$. By the Souslin Theorem [11, 14.2], the Borel set $K_{C}\cap f_{C}(\tilde{C})$ is a continuous image of $\omega^{\omega}$. Using thsi fact and the definition of the cardinal $\mathfrak{b}>|Y|\geq|f(C\cap A)|$, we can find a $\sigma$-compact set $\Sigma_{C}\subset K_{C}\cap f_{C}(\tilde{C})$ that contains the set $f(C\cap A)$. Then

[TABLE]

and hence $f_{C}(C\cap A)=\Sigma_{C}\cap Y$. Then $C\cap A=f_{C}^{-1}(\Sigma_{C}\cap Y)=f_{C}^{-1}(\Sigma_{C})\cap X$. The continuity of the map $f_{C}^{-1}:f_{C}(\tilde{C})\to\tilde{C}$ and the inclusion $\Sigma_{C}\subset f_{C}(\tilde{C})$ imply that the preimage $\Lambda_{C}=f_{C}^{-1}(\Sigma_{C})$ is a $\sigma$-compact subset of the Polish space $\tilde{C}\subset\mathbb{R}$ such that $\Lambda_{C}\cap X=f_{C}^{-1}(\Sigma_{C})\cap X=C\cap A$. Then the union $\Lambda=\bigcup_{C\in\mathcal{C}}\Lambda_{C}$ is a $\sigma$-compact set in $\mathbb{R}$ such that

[TABLE]

which means that the set $A$ is of type $F_{\sigma}$ in $X$. But this contradicts the choice of the set $A$. This contradiction shows that one of the maps $f$ or $f^{-1}$ is not $\sigma$-continuous and hence $\mathfrak{c}_{\sigma}\leq|A|=|B|=\mathfrak{q}_{0}$. ∎

Let us recall that a Polish space is called perfect if it has no isolated points. A subset $A$ of a Polish space $X$ is called perfectly meager if for any perfect Polish subspace $P\subset X$ the intersection $P\cap A$ is meager in $P$. By a result of Grzegorek [7], [8] (see also [14, 5.4]), the real line $X$ contains a perfectly meager subset $A$ of cardinality $|A|=\mathrm{non}(\mathcal{M})$. By the definition of the cardinal $\mathrm{non}(\mathcal{M})$, there exists a non-meager set $B\subset\mathbb{R}$ of cardinality $|B|=\mathrm{non}(\mathcal{M})$. Since $|B|=\mathrm{non}(\mathcal{M})=|A|$, there exists a bijective map $f:B\to A$. We claim that one of the maps $f$ or $f^{-1}$ is not $\sigma$-continuous. To derive a contradiction, assume that the maps $f$ and $f^{-1}$ are $\sigma$-continuous. Then we can find a countable cover $\mathcal{C}$ of $B$ such that for every $C\in\mathcal{C}$ the restriction $f{\restriction}C$ is a topological embedding. By the Lavrentiev Theorem [11, 3.9], for every $C\in\mathcal{C}$ the topological embedding $f{\restriction}C$ can be extended to a topological embedding $f_{C}:\tilde{C}\to\mathbb{R}$ of some $G_{\delta}$-subset $\tilde{C}$ of $\mathbb{R}$. Let $U_{C}$ be the union of open countable subsets in $\tilde{C}$. The hereditary Lindelöf property of the space $\tilde{C}\subset\mathbb{R}$ implies that the open set $U_{C}$ is countable and hence meager in $\mathbb{R}$. On the other hand, the complement $P_{C}:=\tilde{C}\setminus U_{C}$ is a perfect Polish space. Since $f_{C}$ is a topological embedding, the image $f_{C}(P_{C})$ is a perfect Polish subspace of $\mathbb{R}$. The perfect meagerness of $A$ ensures that the intersection $f_{C}(P_{C})\cap A$ is a meager subset of the Polish space $f_{C}(P_{C})$ and then the preimage $(f_{C}{\restriction}P_{C})^{-1}(A)$ is a meager subset of $P_{C}$ and of $\mathbb{R}$, too. Then the preimage $f_{C}^{-1}(A)\subset U_{C}\cup(f_{C}{\restriction}P_{C})^{-1}(A)$ is a meager subset of $\mathbb{R}$ and so is the set $C=(f_{C}{\restriction}C)^{-1}(A)$. Then the set $B=\bigcup\mathcal{C}$ is meager in $\mathbb{R}$, which contradicts the choice of $B$. This contradiction shows that one of the maps $f$ or $f^{-1}$ is not $\sigma$-continuous and hence $\mathfrak{c}_{\sigma}\leq|A|=|B|=\mathrm{non}(\mathcal{M})$.

A subset $A\subset\mathbb{R}$ has universal measure zero if $\mu(A)=0$ for any Borel continuous probability measure $\mu$ on $\mathbb{R}$. By a result of Grzegorek and Ryll-Nardzewski [9] (see also [14, 5.4]), the real line contains a subset $A$ of universal measure zero that has cardinality $|A|=\mathrm{non}(\mathcal{N})$. By the definition of the cardinal $\mathrm{non}(\mathcal{N})$, there exists a subset $B\notin\mathcal{N}$ of $\mathbb{R}$ with $|B|=\mathrm{non}(\mathcal{N})$. Since $|B|=\mathrm{non}(\mathcal{N})=|A|$, there exists a bijective map $g:B\to A$. We claim that the map $g$ is not $\sigma$-continuous. To derive a contradiction, assume that $g$ is $\sigma$-continuous and find a countable cover $\mathcal{C}$ of $B$ such that for every $C\in\mathcal{C}$ the restriction $f{\restriction}C$ is continuous. Dividing each set $C\in\mathcal{C}$ into countably many pieces, we can assume that $C$ is bounded in the real line.

By the Kuratowski Theorem [11, 3.8], for every $C\in\mathcal{C}$ the continuous map $g{\restriction}C$ can be extended to a continuous map $g_{C}:\tilde{C}\to\mathbb{R}$ defined on some $G_{\delta}$-subset $\tilde{C}\subset\bar{C}$ of $\mathbb{R}$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. For every $C\in\mathcal{C}$ consider the subset $A^{\prime}_{C}=\{a\in A:\lambda(g_{C}^{-1}(a))>0\}$ and observe that it is at most countable. Then the set $A^{\prime}=\bigcup_{C\in\mathcal{C}}A_{C}^{\prime}$ is at most countable, too. Let $P=\bigcup_{C\in\mathcal{C}}g_{C}^{-1}(A^{\prime})$ and $\mathcal{C}^{\prime}=\{C\in\mathcal{C}:\lambda(\tilde{C}\setminus P)>0\}$. For every $C\in\mathcal{C}^{\prime}$ the set $\tilde{C}\setminus P$ has finite non-zero measure. So, we can define a Borel probability measure $\mu_{C}$ on $\mathbb{R}$ letting $\mu_{C}(X)={\lambda(g_{C}^{-1}(X)\setminus P)}/{\lambda(\tilde{C}\setminus P)}$ for any Borel subset $X\subset\mathbb{R}$. It is easy to see that the measure $\mu_{C}$ is continuous (which means that the measure of each singleton is zero). Since the set $A$ has universal measure zero,

[TABLE]

Then $\lambda(B\setminus P)\leq\sum_{C\in\mathcal{C}}\mu(B\cap C\setminus P)=0$ and

[TABLE]

as $B\cap P=g^{-1}(A^{\prime})$ is at most countable. But the equality $\lambda(B)=0$ contradicts the choice of $B$. This contradiction shows that the map $g$ is not $\sigma$-continuous and hence $\mathfrak{c}_{\sigma}\leq|B|=\mathrm{non}(\mathcal{N})$. ∎

[TABLE]

is consistent with ZFC?

For a cardinal $\kappa$ let $\mathfrak{b}_{\kappa}$ be the smallest cardinality of a subset $A$ of a $G_{<\kappa}$-set $G$ in $\mathbb{R}$ such that $A$ is not contained in a $\sigma$-compact subset of $G$. It can be shown that $\mathfrak{b}_{\kappa}$ is a non-increasing function on the variable $\kappa$ such that

[TABLE]

By [19], for every $A\in\mathcal{I}_{P}$ the image $P(A)$ is a meager subset of $\omega^{\omega}$. This implies that $\mathrm{non}(\mathcal{I}_{P})\leq\mathrm{non}(\mathcal{M})$ and $\mathrm{cov}(\mathcal{I}_{P})\geq\mathrm{cov}(\mathcal{M})$. By [19], it is consistent that $\mathrm{cov}(\mathcal{I}_{P})>\mathrm{cov}(\mathcal{M})$.