Last time, we embarked on proving that the projective limit of a projective system of compact sober (resp., and non-empty) spaces is compact and sober (resp., and non-empty), a theorem that Fujiwara and Kato call Steenrod’s Theorem. However, instead, we merely proved that a projective limit of a projective system of non-empty compact sober spaces is non-empty. Do not despair: this is the essential argument in the proof of Steenrod’s Theorem, which we complete this month. Read the full post.