What is the meaning of this statement in EGA I (2.2.4) This is a statement made in EGA I (2.2.4) "Tout espace noetherien est quasi-compact;inversement, tout espace topologique dans lequel tout ouvert est quasi-compact est noetherien." The first statement is clear from ascending chain property of open sets. However, I do not get the 2nd statement. $\textbf{Q:}$ Is the second statement saying that if $X$ is a topological space s.t. any $U\subset X$ open is quasi compact, then $X$ is noetherian? This does not look obvious at all and it looks very strange. If $X$ has finite covering property and $X$ is locally noetherian, then it follows easily that $X$ is noetherian. Finite covering property is guaranteed by all open sets compact. However, $X$ local noetherian property is left out here. What is the translation of the 2nd statement? Have I misunderstood the statement? In other words, I can conclude that $X$ is noetherian iff $X$ is locally noetherian and $X$ is quasi-compact.

There is a community translation project for EGA at < they translate your sentence as "(2.2.4). Any Noetherian space is quasi-compact; conversely, any topological space in which all open sets are quasi-compact is Noetherian." so it sounds like you have the right translation. And there are some remarks on the proof at this question Are there non-noetherian topological spaces in which every open subset is quasi-compact?