On d-properties of topological spaces (Q753189)

The author introduces new properties of spaces called d-paracompactness, d-para-Lindelöfness, d-metacompactness, d-meta-Lindelöfness, od- metacompactness and od-meta-Lindelöfness. Then he obtains a series of interesting results concerning those properties. A covering \({\mathcal U}\) of a space X is said to be d-locally finite (resp. countable) if the set of all points at which \({\mathcal U}\) is locally finite (resp. countable) is a dense subset of X. Further, \({\mathcal U}\) is said to be d-pointwise finite (resp. countable) if there is a dense subset D of X such that each point of D belongs to at most finitely (resp. countably) many members of \({\mathcal U}\). If D can be chosen to be an open and dense subset of X, then \({\mathcal U}\) is said to be od-pointwise finite (resp. countable). A space X is d-paracompact (resp. d-para-Lindelöf, d-metacompact, d- meta-Lindelöf, od-metacompact, od-para-Lindelöf) if each open covering of X has a d-locally finite (resp. d-locally countable, d- pointwise countable, od-pointwise finite, od-pointwise countable) open refinement.