Lemma 5.12.13. Let $X$ be a Noetherian topological space.

1. The space $X$ is quasi-compact.

2. Any subset of $X$ is retrocompact.

Proof. Suppose $X = \bigcup U_ i$ is an open covering of $X$ indexed by the set $I$ which does not have a refinement by a finite open covering. Choose $i_1, i_2, \ldots$ elements of $I$ inductively in the following way: Choose $i_{n + 1}$ such that $U_{i_{n + 1}}$ is not contained in $U_{i_1} \cup \ldots \cup U_{i_ n}$. Thus we see that $X \supset (X \setminus U_{i_1}) \supset (X \setminus U_{i_1} \cup U_{i_2}) \supset \ldots$ is a strictly decreasing infinite sequence of closed subsets. This contradicts the fact that $X$ is Noetherian. This proves the first assertion. The second assertion is now clear since every subset of $X$ is Noetherian by Lemma 5.9.2. $\square$

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