# Stable subsets with respect to pointwise convergence.

Consider the linear spacet $\mathcal{F}(\mathbb{R}^n)$ of all real functions defined in $\mathbb{R}^n$. It is well known that the subspace $\mathcal{C}(\mathbb{R}^n)$ of all real valued continuous function defined in $\mathbb{R}^n$ is stable with respect to the uniform (convergence) limit of elements in $\mathcal{C}(\mathbb{R}^n)$.

Question 1: Which is the smallest set (with respect to inclusion relation) containing $\mathcal{C}(\mathbb{R}^n)$ and stable with respect to pointwise convergence?

Question 2: Which is the smallest linear subspace of $\mathcal{F}(\mathbb{R}^n)$ which is stable with respect to pointwise convergence?

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These are the Baire functions. –  Andreas Blass Apr 30 '12 at 0:33
Andreas's answer is for question 1; $\lbrace0\rbrace$ will do for question 2. –  François G. Dorais Apr 30 '12 at 11:11
Thankyou very much. My question started from a guess between the equivalence of "what I now know to be the class of Baire functions" and Borel functions. Dut to your answers I was able to find this www.jstor.org/stable/1996801. So thanks again. –  Josh Apr 30 '12 at 12:21