The D-property of some Lindelِf spaces and related conclusions

It is shown that the space C p ( τ ω ) is a D-space for any ordinal number τ, where τ ω = { α ⩽ τ : cf ( α ) ⩽ ω } . This conclusion gives a positive answer to R.Z. Buzyakovaʹs question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshinaʹs conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.