We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w\times 2^w$ and continuous functions $e,f:w^w\to w^w$ such that  • N is $G_{\delta }$ and $N_x:x\in w^w$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_{\sigma }$ and $M_x:x\in w^w$ is a basis for the ideal of meager subsets of $2^w$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq N_y\Rightarrow M_x\subseteq M_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_{\delta }$) sets $B\subseteq X\times 2^w$ with all...

Three sets occurring in functional analysis are shown to be of class PCA (also called ${\Sigma }_2^1$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

We show that if $X$ is a ${\Sigma }_1^1$ separable metrizable space which is not $\sigma$-compact then $C_p^{*}\left(X\right)$, the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_1^1$-complete. Assuming projective determinacy we show that if $X$ is projective not $\sigma$-compact and $n$ is least such that $X$ is ${\Sigma }_n^1$ then $C_p\left(X\right)$, the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_n^1$-complete. We also prove a simultaneous improvement of theorems of Christensen...

Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f: X → Y be a continuous function onto Y ⊂ C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions.