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A neighbourhood assignment in a space X is a family O={Ox:x∈X} of open subsets of X such that x∈Ox for any x∈X. A set Y⊆X is a kernel of O if O(Y)=⋃{Ox:x∈Y}=X. We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some...
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