The polynomial algebra is dense but not closed on a nondegenerate compact interval

Given: Reals $a<b$ and the algebra $P[a,b]$ of restricted real polynomials.

Every unital point-separating real function algebra on a compact Hausdorff space is uniformly dense in the full real continuous-function space (Real Stone–Weierstrass theorem for compact Hausdorff spaces).

For $a\le b$, every continuous real function on $[a,b]$ is a uniform limit of polynomials (Polynomials are uniformly dense in $C([a,b],\mathbb{R})$ for every closed interval).

A nonzero real polynomial of degree $n$ has at most $n$ distinct real roots (A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots).

For $a\le b$, every family of open subsets of $\mathbb{R}$ whose union contains $[a,b]$ has a finite subfamily whose union already contains $[a,b]$ (Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact).

The function $d_{\mathbb{R}}(s,t)=\mathrm{\mid }s-t\mathrm{\mid }$ is a metric on $\mathbb{R}$, and its metric topology is the usual topology (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

Every metric space is Hausdorff: distinct points are separated by disjoint open balls (Distinct points of a metric space have disjoint balls around them).

A subset $A$ of a topological space $X$ is a compact subset — that is, the subspace $(A,{\mathcal{T}}_A)$ is a compact space — if and only if every family of open subsets of $X$ whose union contains $A$ has a finite subfamily whose union contains $A$, or else $A=\mathrm{\varnothing }$ (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, clause 1).

Hausdorffness is hereditary: every subspace of a Hausdorff space is Hausdorff ($T_0$, $T_1$, and Hausdorffness are hereditary, Hereditary, open-hereditary and closed-hereditary properties of topological spaces).

The inclusion of a subspace into its ambient space is continuous (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

For continuous $f,g:X\to \mathbb{R}$ on a topological space, $f+g$, $fg$ and $\mathrm{\mid }f\mathrm{\mid }$ are continuous (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined).

A map is continuous when the preimage of every open set containing an image point contains an open set around that point (Continuity of a map of topological spaces at a point and globally).