Equivalent definitions of Noetherian topological space
Let $R$ be a commutative ring. It is known that $X=\operatorname{Spec}(R)$ is noetherian iff $R$ satisfies ACC on radical ideals.
If $R$ is a valuation ring, then every proper radical ideal is prime; see Geraschenko, Commutative Rings, Theorem 30.5. Thus $X=\operatorname{Spec}(R)$ satisfies the condition $1')$. Furthermore, in this case $X=\operatorname{Spec}(R)$ is noetherian iff $R$ satisfies ACC on prime ideals. Now take $R$ a valuation ring which has an infinite chain of prime ideals.