## 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.

### MSC:

 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