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D-properties of Finite unions of spaces with point countable weak bases and satisfying open(G)
- GUO Hong-feng, LI Yu-si, SUN Wei-hua
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JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(10):
72-76.
doi:10.6040/j.issn.1671-9352.0.2016.571
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The relation between point-countable weak bases and D-property is studied. It is shown that, if a space X of countable tightness is the union of finitely many subspaces Xi with point-countable weak base Ti={Ti(x):x∈Xi} satisfying Ti(x)∩Ti(y)=Ø for any distinct x,y∈X, then X is a D-space. And then the relation is studied between open(G)and D-property. We obtain that, if X=X1∪X2, where both X1 and X2 satisfy open(G), then X1^-∩X2^- satisfies open(G). With the help of this result, a detailed proof is shown at last for the result that the union of finitely many subspaces satisfying open(G)is a D-space.