
Re: Property related to denseness
Posted:
Jan 12, 2013 7:27 PM


On Jan 12, 5:41 pm, Paul <pepste...@gmail.com> wrote: > Let A be a subset of the topological space of X. > What is the standard terminology for the property that X = the intersection of all the open sets that contain A?
I don't know, this is the first I've heard of it.
> This seems related to A being dense in X but it is clearly not the same. > Is this a stronger property than denseness? > Where does this property stand in relation to denseness?
A dense set does not necessarily have your property. If X has the T_1 property (all oneelement subsets are closed), then the only subset with your property is X itselt. In fact, if X contains a point x such that {x} is closed but not open, then X \ {x} is dense in X but does not have your property.
A set with your property is not necessarily dense. (Of course the counterexample can't be a T_1space.) Let X be the closed interval [0, 1] with the topology consisting of all initial segments of [0, 1], that is, the closed intervals [0, x], 0 <= x <= 1, the halfopen intervals [0, x), 0 < x <= 1, and the empty set. The set {1} is not dense (being closed), but it has your property.

