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Let $X$ be a Noetherian compact $T_0$ topological space such that every closed subset of $X$ is locally connected. Is there any characterization for such a space? Any hint is appreciated.

Recall that a topological space $X$ is called Noetherian if any ascending chain of open subsets stabilizes after finitely many steps, equivalently, any non-empty set of closed subsets of $X$, ordered by inclusion, has a minimal element.

Let $X$ be a compact $T_0$ topological space such that every closed subset of $X$ is locally connected. Is there any characterization for such a space? I guess $X$ is Noetherian, but I cannot prove that. Any hint is appreciated.

Recall that a topological space $X$ is called Noetherian if any ascending chain of open subsets stabilizes after finitely many steps, equivalently, any non-empty set of closed subsets of $X$, ordered by inclusion, has a minimal element.

Let $X$ be a compact $T_0$ space such that every closed subset of $X$ is locally connected. Is there any characterization for such a space? Any hint is appreciated.

Let $X$ be a Noetherian compact $T_0$ topological space such that every closed subset of $X$ is locally connected. Is there any characterization for such a space? Any hint is appreciated.

Recall that a topological space $X$ is called Noetherian if any ascending chain of open subsets stabilizes after finitely many steps, equivalently, any non-empty set of closed subsets of $X$, ordered by inclusion, has a minimal element.