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?