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

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Property related to denseness
Posted: Jan 14, 2013 2:15 PM
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On Jan 14, 12:55 pm, 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 = 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 = 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. By "the trivial topology" I guess you mean the
"indiscrete" topology, in which the only open sets are the empty set
and X itself? If X has the indiscrete topology, then every nonempty
subset A will have the OP's property. On the other hand, if X is a
topological space in which all one-point sets are closed (a so-called
T_1-space), then the OP's condition holds only for A = X.



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