As usual $C\left(X\right)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$. It is well-known that if $X$ and $Y$ are realcompact spaces such that $C\left(X\right)$ and $C\left(Y\right)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C\left(X\right)$ $X$. The restriction to realcompact spaces stems from the fact that $C\left(X\right)$ and $C\left(\upsilon X\right)$ are isomorphic, where $\upsilon X$ is the (Hewitt) realcompactification of $X$. In this note, a class of locally compact spaces $X$ that includes properly the class of locally...

Let $(X,𝕀)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R\left(A^2\right)$ is completely $𝕀$-nonmeasurable, i.e, it is $𝕀$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations ${\left(R_{\alpha }\right)}_{\alpha <2^{\omega }}.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.

We give a classical proof of the theorem stating that the $\sigma$-ideal of meager sets is the unique $\sigma$-ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

We calculate the cardinal characteristics of the $\sigma$-ideal $\mathscr{H}𝒩\left(G\right)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov\left(\mathscr{H}𝒩\right(G\left)\right)\le 𝔟\le max\{𝔡,non(𝒩\left)\right\}\le non\left(\mathscr{H}𝒩\right(G\left)\right)\le cof\left(\mathscr{H}𝒩\right(G\left)\right)>min\{𝔡,non(𝒩\left)\right\}$. If $G={\prod }_{n\ge 0}{G}_n$ is the product of abelian locally compact groups $G_n$, then $add\left(\mathscr{H}𝒩\right(G\left)\right)=add\left(𝒩\right)$, $cov\left(\mathscr{H}𝒩\right(G\left)\right)=min\{𝔟,cov(𝒩\left)\right\}$, $non\left(\mathscr{H}𝒩\right(G\left)\right)=max\{𝔡,non(𝒩\left)\right\}$ and $cof\left(\mathscr{H}𝒩\right(G\left)\right)\ge cof\left(𝒩\right)$, where $𝒩$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof\left(\mathscr{H}𝒩\left(G\right)\right)>2^{{\aleph }_0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a...