
Re: Property related to denseness
Posted:
Jan 14, 2013 2:15 PM


On Jan 14, 12:55 pm, mstem...@walkabout.empros.com (Michael Stemper) wrote: > In article <d2c170e4b59b4c738a7324374fc8b6e1@googlegroups.com>, Paul <pepste...@gmail.com> writes: > >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? > > The trivial topology? > > If these two Xs refer to the same thing, then I don't see how X could be > the intersection of more than one subset of X, and I don't see how that > subset could be anything other than X.
Yes, the OP's property that "X = the intersection of all the open sets that contain A" could be stated more simply as "X is the only open set that contains A". This is, of course, a property of a subset A of a topological space X. By "the trivial topology" I guess you mean the "indiscrete" topology, in which the only open sets are the empty set and X itself? If X has the indiscrete topology, then every nonempty subset A will have the OP's property. On the other hand, if X is a topological space in which all onepoint sets are closed (a socalled T_1space), then the OP's condition holds only for A = X.

