Notation related to the Borel hierarchy

Question

I'm reading an article where the author uses many classes in the Borel hierarchy: namely, $\mathcal F_\sigma$, $G_\delta$, $\mathcal F_{\sigma\delta}$, etc. In this context, he mentions the class $\mathcal F\cap\mathcal G$ (no subscripts). Does anyone knows what this notation means?

Answer 1

The pointclass $\mathcal{F\cap G}$ usually denotes the collection of clopen sets, at least in Hausdorff's convention going back to his Grundzüge der Mengenlehre (1914). Hausdorff defined topological spaces in terms of neighborhood bases and he wrote

$$\mathcal{F} = \{F \subseteq X \mid F \text{ is closed}\},\quad \mathcal{G} = \{G \subseteq X \mid G \text{ is open}\}$$

for the corresponding collections of closed sets and open sets. Here $F$ is for French fermé (closed) and $G$ is for German Gebiet ((open) neighborhood). Thus, $$\mathcal{F \cap G} = \{H \subseteq X \mid H \text{ is closed and open}\}.$$

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For every collection $\mathcal{H} \subseteq \mathcal{P}(X)$ Hausdorff defined $$ \mathcal{H}_\sigma = \left\{A \subseteq X \mid \exists H_n \in \mathcal{H}, n \in \mathbb{N} : A = \bigcup\nolimits_{n\in \mathbb{N}} H_n\right\} $$ and $$ \mathcal{H}_\delta = \left\{A \subseteq X \mid \exists H_n \in \mathcal{H}, n \in \mathbb{N} : A = \bigcap\nolimits_{n\in \mathbb{N}} H_n\right\}. $$ Here $\sigma$ is the small Greek s for German Summe (sum, old term/notation for union) and $\delta$ is the small Greek d for German Durchschnitt (intersection). If I remember correctly, he also used $\mathcal{H}_s$ and $\mathcal{H}_d$ for finite unions and intersections from $\mathcal{H}$ which partly explains the usage of Greek letters.

Applying this to the classes $\mathcal{H} = \mathcal{F,G,F_\sigma,G_\delta}, \dots$, you recover the usual pointclasses in the Borel hierarchy.