Definition. Let $C$ be an algebraically closed field containing $k$. If $S$ is a subset of $k[x_{1},...,x_{n}]$, then the zero set of $S$ is $$Z(S) = \lbrace (a_{1},...,a_{n}): f(a_{1},...,a_{n}) = 0\;\mathrm{for\,all}\;f \in S\rbrace$$
The sets $\lbrace Z(S):S \subseteq k[x_{1},...,x_{n}] \rbrace$ are closed sets of a topology on $C^{n}$ (Zariski Topology).
We know that the Hilbert Basis Theorem implies $k[x_{1},...,x_{n}]$ is a Noetherian ring, so $k[x_{1},...,x_{n}]$ satisfies the accending condition chain.
Question. How can I show, using the Hilbert Basis Theorem, that $C^{n}$ is a Noetherian space?
I proved in a previous question that the following statements are equivalent:
(i) The space $V$ is Noetherian space.
(ii) Any nonempty collection $\lbrace U_{\alpha}\rbrace$ of open subsets of $V$ has a maximal element; that is, there is a $U \in \lbrace U_{\alpha}\rbrace$ not properly contained in any other element of $\lbrace U_{\alpha}\rbrace$.
(iii) The space $V$ satisfies the descending chain condition on closed sets: If $C_{1} \supseteq C_{2} \supseteq ...$ is a decreasing chain of closed subsets of $V$, then there is an $n$ with $C_{n} = C_{n+r}$ for each $r \geq1$.
Also, I proved that:
If $V$ is a Noetherian topological space, then $V$ can be written as a finite union of closed irreducible subsets.
Strong and Weak Hilbert Nullstellensatz.
I wanted, necessarly, to use the Hilbert Basis Theorem.Can anybody help me with a hint?
For any subset $W \subset k^n$ we can construct the ideal $$I(W) = \{f \in k[x_1, \dots, x_n] : f(W) = 0\}$$ of polynomials that vanish on $W$. Observe that $Z(I(Z(S))) = Z(S)$ for all $S$ -- in other words, $Z(I(W)) = W$ for all closed $W$. Given a chain $W_1 \supseteq W_2 \supseteq W_3 \supseteq \dots$ of closed subsets one obtains an increasing chain $I(W_1) \subseteq I(W_2) \subseteq \dots$ of ideals. Now you can use Hilbert's Basis Theorem.