  Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye, we prove
that the maximal $\infty$-step pro-nilfactor $X_\infty$ of a minimal system
$(X,T)$ is the topological characteristic factor along polynomials in a certain
sense. Namely, we show that by an almost one to one modification of $\pi:X\to
X_\infty$, the induced open extension $\pi^*:X^*\to X_\infty^*$ has the
following property: for any $d\in \mathbb{N}$, any open subsets
$V_0,V_1,\ldots,V_d$ of $X^*$ with $\bigcap_{i=0}^d \pi^*(V_i)\neq \emptyset$
and any distinct non-constant integer polynomials $p_i$ with $p_i(0)=0$ for
$i=1,\ldots,d$, there exists some $n\in \mathbb{Z}$ such that $V_0\cap
T^{-p_1(n)}V_1\cap \ldots \cap T^{-p_d(n)}V_d \neq \emptyset$. where an integer
polynomial is the polynomial with rational coefficients taking integer values
on the integers.
  As an application, the following result is obtained: for a totally minimal
system $(X,T)$ and integer polynomials $p_1,\ldots,p_d$, if every non-trivial
integer combination of $p_1,\ldots,p_d$ is not constant, then there is a dense
$G_\delta$ subset $\Omega$ of $ X$ such that the set \[ \{(T^{p_1(n)}x,\ldots,
T^{p_d(n)}x):n\in \mathbb{Z}\} \] is dense in $X^d$ for every $x\in \Omega$.
