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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 12, 2013 7:27 PM
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On Jan 12, 5:41 pm, Paul <pepste...@gmail.com> wrote:
> 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?


I don't know, this is the first I've heard of it.

> This seems related to A being dense in X but it is clearly not the same.
> Is this a stronger property than denseness?
> Where does this property stand in relation to denseness?


A dense set does not necessarily have your property. If X has the T_1
property (all one-element subsets are closed), then the only subset
with your property is X itselt. In fact, if X contains a point x such
that {x} is closed but not open, then X \ {x} is dense in X but does
not have your property.

A set with your property is not necessarily dense. (Of course the
counterexample can't be a T_1-space.) Let X be the closed interval [0,
1] with the topology consisting of all initial segments of [0, 1],
that is, the closed intervals [0, x], 0 <= x <= 1, the half-open
intervals [0, x), 0 < x <= 1, and the empty set. The set {1} is not
dense (being closed), but it has your property.



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