Re: Topology question

From: Gabriel Chime (g_chime_at_yahoo.com)
Date: 08/11/04 

Date: 11 Aug 2004 13:24:27 -0700

"BCAL" <bca1@xs4all.nl> wrote in message news:<4119f75b$0$10528$e4fe514c@news.xs4all.nl>...
> --
>
>
> Gabriel Chime <g_chime@yahoo.com> wrote in message
> news:48349cf7.0408102317.3506ed31@posting.google.com...
> > I'm reading a book on real analysis and came across the following:
> > Every open set or real numbers is the union
> > of a countable collection of disjoint open intervals.
> Indeed, these are exactly the connected components of the open set.
> Mind you, we have to consider R and sets of the form {x : x < a} and {x : x
> > a} as open intervals as well, say
> with + or -infinity as "delimiter".
>
> >
> > Is this property unique to R or does it generalize to other spaces?
> One needs an order topology for this (eg. R^2 is not orderable (in the
> topological sense), so it makes no sense to
> talk about "intervals".

How will the above property fail if we use 2
dimensional intervals
(cross products of intervals in R)?

Going back to my original question, a topology with a countable
base would be a generalization, right?

> But if X is an ordered space, this property would hold in a locally
> connected and ccc (any family of disjoint
> non-empty open subsets is at most countable; this is implied by having a
> countable dense subset) ordered topological space.
>
> > Is there a name for this property?
>
> Not as far as I know.
>
> > Thanks in advance.
>
> Henno

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