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



quasi
Posts:
12,067
Registered:
7/15/05


Re: Property related to denseness
Posted:
Jan 16, 2013 6:17 AM


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



