# Are there non-noetherian topological spaces in which every open subset is quasi-compact?

I'm working through an exercise in Hartschorne concerning direct limits of sheaves on noetherian topological spaces. It seems natural to use the the fact that every open subset of a noetherian topological space is the quasi-compact in its induced topology. I'm some trouble proving what I need when I only use this property. I'm guessing that I'll have to use noetherian-ness more directly, but I thought I would ask if anyone knew an example offhand.

Is there a non-notherian topological space which has this property?

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