# Is this observation about the Borel Hierarchy trivial?

Hello, consider the following theorem. Is it trivial? Is it interesting? Is it worth including in a paper if I can prove it in 1 line as a corollary?

Theorem: Suppose $n>0$ is a natural. Suppose $S=\cup_{i\in\mathbb{N}}\cap_{j\in\mathbb{N}}X_{ij}= \cap_{i\in\mathbb{N}}\cup_{j\in\mathbb{N}}Y_{ij}$, where the $X_{ij},Y_{ij}$ are $\Delta_n^0$ subsets of Baire space. Then there exists a single family $Z_{ij}$ of $\Delta_n^0$ subsets of Baire space such that $S=\cup_{i\in\mathbb{N}}\cap_{j\in\mathbb{N}}Z_{ij}=\cap_{i\in\mathbb{N}}\cup_{j\in\mathbb{N}}Z_{ij}$.

Edit: Forgot to mention, I'm talking about boldface pointclasses here.

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Aren't $\Delta^0_n$ sets closed under countable unions and intersections? –  Andres Caicedo May 29 '11 at 22:35
@Andres: No. $\Delta^0_1$ (boldface) is clopen. –  Andreas Blass May 29 '11 at 23:13
Andreas: Oh, of course. –  Andres Caicedo May 29 '11 at 23:16
Well, I ended up adding it. If anybody wants to see the proof, here is the paper, the corollary is at the very end: math.osu.edu/~alexander/lopsided.pdf –  Sam Alexander Jun 4 '11 at 16:00