Let $(\mathbb{R}^{\mathbb{R}}, p)$ be the space of all functions from $\mathbb{R}$ to $\mathbb{R}$ with topology of Pointwise convergence. I need to find closure of set of all polynomials without constant term in $\mathbb{R}^{\mathbb{R}}$.
I dont know how to approach to this problem. Hints?
You exactly get the set $\{f: \mathbb{R} \to \mathbb{R}: f(0) = 0\}$, which is pointwise closed as it equals $\pi_0^{-1}[\{0\}]$.
To prove this, think about why the set of all polynomials is dense in your space: finitely many input-output pairs determine a polynomial.