A classical result from Lusin and Souslin tells us that should a Borel function $f:X\to Y$ be a bijection, then its inverse $f^{-1}:Y\to X$ must also be a Borel function. Let $bB=\{f:[0,1]-\mathbb{Q}\to [0,1]-\mathbb{Q}:f\text{is Borel measurable}\}$. Here, we consider the sets $b{B}^{\prime }=\{f\in bB:f$ is one-to-one}, and $bH=\{f\in b{B}^{\prime }:f:[0,1]-\mathbb{Q}\to [0,1]-\mathbb{Q}\text{is a bijection}\}$, each endowed with the metric $d(f,g)={\sup }_{x\in [0,1]-\mathbb{Q}}|f(x)-g(x)|$, and the latter also with the metric $d^{⁎}(f,g)=d(f,g)+d(f^{-1},g^{-1})$. Let $\alpha <\mathrm{\Omega }$. Then

1.

the set $b{B}_{\alpha }^{{}^{\prime }}=\{f\in b{B}^{\prime }:f\text{is Borel-}\alpha \}$ is nowhere dense and perfect in $(b{B}^{\prime },d)$, and for any $\beta <\mathrm{\Omega }$, there exists $\mathcal{D}$ a dense subset of $(b{B}_{\alpha }^{\prime },d)$ such that $f^{-1}\in b{B}_{\beta }$, for any $f\in \mathcal{D}$,

2.

the set $b{H}_{\alpha }=\{f\in bH:f$ is Borel-α} is nowhere dense and perfect in $(bH,d)$, and for any $\beta <\mathrm{\Omega }$ a successor ordinal, there exists $\mathcal{D}$ a dense subset of $(b{H}_{\alpha +1},d)$ such that $f^{-1}\in b{H}_{\beta }$, for any $f\in \mathcal{D}$, and

3.

the set $(b{H}_{\alpha },d^{⁎})$ is closed and nowhere dense in $(bH,d^{⁎})$, and for any $\beta <\mathrm{\Omega }$, the set $H(\alpha ,\beta )=\{f\in b{H}_{\alpha }:f^{-1}\in b{H}_{\beta }\}$ is closed and nowhere dense in $(bH,d^{⁎})$.