Borel $\boldsymbol{\Delta}^0_\alpha$ vs. $\bigcup_{\xi<\alpha}\boldsymbol{\Delta}^0_{\xi}$ for limit $\alpha$

Question

It's clear for limit $\alpha$ that $\bigcup_{\xi<\alpha}\boldsymbol{\Sigma}^0_{\xi}\subsetneq \boldsymbol{\Sigma}^0_{\alpha}$, but is this obviously true for the ambiguous pointclasses? Are there any simple examples of elements in $\boldsymbol{\Delta}^0_\alpha\setminus\bigcup_{\xi<\alpha}\boldsymbol{\Delta}^0_\xi$ for limit $\alpha$? I'm probably just missing something obvious since I'm more used to thinking about $\alpha<\omega$.

Answer 1

Basically the same idea works for the ambiguous pointclasses: we just take an appropriately "separated" union of increasingly complex sets.

Specifically, fix $\alpha$ a countable limit ordinal, let $(\beta_i)_{i\in\omega}$ be a cofinal sequence in $\alpha$, and for each $i\in\omega$ let $X_i\subseteq [i,i+1)$ be in $\bf \Delta^0_{\beta_i+1}\setminus \Delta^0_\beta$. Then $Y=\bigcup_{i\in\omega}X_i$ is $\bf\Delta^0_\alpha$ but not $\bf\Delta^0_\beta$ for any $\beta<\alpha$.