A lemma of Cielsielski

Reading section 2 of this paper '' A century of Sierpinski–Zygmund functions'' (Ciesielski, K.C., Seoane-Sepúlveda, J.B.. Rev. R. Acad.Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (2019)), i wonder if some conditions requested were really necessary.

For example to page 6 why i esclude rational point from $P_f$? ($P:=P_f\setminus\mathbb{Q}$) and why considered a countable basis of $\mathbb{R}$ of nonempty intervals with rationalendpoints. I don't find it useful to talk about rational numbers in this case.

Most important: in first part of lemma 2.3 what are the sets $\hat U$ and $\hat V$ for?I have proof that if i considered $x\in P\cap \hat B$ and $\mathcal{V}=\mathcal{U}\cup\{U_0\times V:V\in\mathcal{W}\}$ ($\mathcal{V}=\mathcal{U}$ if $\mathcal{W}=\emptyset$) then are satisfied the above properties b),c) and $Y\cap B\neq \emptyset$.

I have difficulty verifying property a); can someone help me?

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