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Topic: Property related to denseness
Replies: 8   Last Post: Jan 16, 2013 4:34 PM

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Posts: 11,311
Registered: 7/15/05
Re: Property related to denseness
Posted: Jan 16, 2013 6:17 AM
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pepstein5@gmail.com wrote:
>Michael Stemper wrote:
>>Butch Malahide writes:
>> >Michael Stemper) wrote:
>> >>Paul <pepste...@gmail.com> writes:
>> >>
>> >> The trivial topology?

>> >
>> > By "the trivial topology" I guess you mean the
>> >
>> >"indiscrete" topology,
>> >

>> Willard also uses that term.

Yes, Willard calls it "the trivial topology".

>>It makes sense that if the finest topology is called
>>"discrete" that the coarsest could be called "indiscrete".

>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.

I think that in this case, the intent of the terminology is that,
for a given set X, the trivial topology on X is the simplest
possible topology on X.

However it's also true that any topology on X is a superset
of the trivial topology on X.

>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.

No -- the terminology "the trivial topology" always refers to
a topology on a given set X.

>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".

Yes, exactly.

>Is "trivial topology on X" a standard way of referring to the
>indiscrete topology on X? If not, I think it should be.

The terminologies

"the trivial topology on X"

"the indiscrete topology on X"

are both accepted, but I think "the indiscrete topology" is the
one more commonly used.

Willard mentions both but favors "the trivial topology".

Wikipedia has "the trivial topology" first


but indicates that "the indiscrete topology" is a perfectly
acceptable alternate.

Personally, I like "the trivial topology" better, but it's not
a big deal either way.


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