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A Noetherian topological space (https://en.wikipedia.org/wiki/Noetherian_topological_space) is a topological space in which any non-empty collection of open sets has a maximal element w.r.t. inclusion i.e. any non-empty collection of closed sets has a minimal element w.r.t. inclusion . It can be shown that every subset of a Noetherian topological space is compact and again Noetherian and also that any closed subset of a Noetherian topological space has only finitely many connected components.

Now let $X$ be a Noetherian topological space and $Y$ be a closed subset of $X$, then does $Y$ have at most as many connected components as that of $X$ ?

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It seems not. Let $X = \{0, 1, 2\}$ with the topology generated by sets $\{0, 1\}$, $\{0, 2\}$. This is a connected noetherian topological space. But $Y := \{1, 2\}$ is closed discrete, and so has two components.

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