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Paul
Posts:
720
Registered:
7/12/10


Re: Property related to denseness
Posted:
Jan 16, 2013 5:13 AM


On Tuesday, January 15, 2013 6:53:50 PM UTC, Michael Stemper wrote: > In article <a1044dfe77d845e89f6f307894d02c86@u19g2000yqj.googlegroups.com>, Butch Malahide <fred.galvin@gmail.com> writes: > > >On Jan 14, 12:55=A0pm, 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 =3D 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 =3D 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. > > > > Okay, thanks for validating my thinking. > > > > > By "the trivial topology" I guess you mean the > > >"indiscrete" topology, > > > > Willard also uses that term. It makes sense that if the finest topology > > is called "discrete" that the coarsest could be called "indiscrete". > >
Not just Willard but "indiscrete" is the standard term I would think. I think that a trivial object would be one that is embedded in all objects of the same type. For example, the trivial group is the group with one element. If there is such a thing as "the trivial topology [without mentioning the underlying set]" then that might be the topology on the empty set where the only open set is the empty set. If the underlying set is X, then I would think "The trivial topology on X" is a fine way of describing the indiscrete topology, since the open sets in that topology on X are exactly the sets that are open in every topology on X. So it's analogous to the "trivial group". Is "trivial topology on X" a standard way of referring to the indiscrete topology on X? If not, I think it should be.
Paul Epstein



