Closures of discrete sets often reflect global properties. (English) Zbl 1002.54021

This is an interesting paper which considers the general question of if the closure of every discrete subset of a space has a particular property, does the whole space then have the property. The positive results include the properties: perfect normality, tightness at most \(\kappa\), and even scattered. Another interesting result is the fact that it is consistent and independent that each of “metrizability” and “second countable” may be such discrete reflective properties. A nice sample open problem raised by the authors gives more of the flavor: if a space \(X\) has the property that the closure of every discrete subset is \(\sigma\)-compact, must \(X\) be Lindelöf.


54H11 Topological groups (topological aspects)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D05 Connected and locally connected spaces (general aspects)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54C25 Embedding
22A05 Structure of general topological groups