Math Forum Discussions

Paul

Posts: 780
Registered: 7/12/10

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

Plain Text

On Tuesday, January 15, 2013 6:53:50 PM UTC, Michael Stemper wrote:
> In article <a1044dfe-77d8-45e8-9f6f-307894d02c86@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 <d2c170e4-b59b-4c73-8a73-24374fc8b6e1@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?
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> >>
>
> >> The trivial topology?
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> >>
>
> >> If these two Xs refer to the same thing, then I don't see how X could be
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> >> the intersection of more than one subset of X, and I don't see how that
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> >> subset could be anything other than X.
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> >
>
> >Yes, the OP's property that "X =3D the intersection of all the open sets
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> >that contain A" could be stated more simply as "X is the only open set
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> >that contains A". This is, of course, a property of a subset A of a
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> >topological space X.
>
>
>
> Okay, thanks for validating my thinking.
>
>
>
> > By "the trivial topology" I guess you mean the
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> >"indiscrete" topology,
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>
>
> Willard also uses that term. It makes sense that if the finest topology
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> 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