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

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quasi

Posts: 10,193
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

<http://en.wikipedia.org/wiki/Trivial_topology>

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.

quasi



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