Re: Compact connected Hausdorff

On Sun, 4 Dec 2005, Narcoleptic Insomniac wrote:
> On Dec 4, 2005 7:03 AM CT, Narcoleptic Insomniac wrote:
> > On Dec 4, 2005 5:02 AM CT, William Elliot wrote:
> >
> > > > Lemma: If space X is compact and connected. For
> > > > any open set U of X, the closure of some component
> > > > of U intersect X - U.
> > >
> > > I'm of the mind that "of some component" was to be
> > > read as "of any component" and that Hausdorff is
> > > tacit assumption.
> > >
> > > Is the surmise correct? Anyway, let C be a component
> > > of open U and assume the negation, that
> > > cl C /\ (X - U) is empty,
> > > ie
> > > (cl C) - U = nulset
> > >
> > > From that comes
> > > C = cl C proper subset U.
> >
> > Since X is compact there exists an open cover K with
> > a finite subcover; that is X = \/ K_i for some finite
> > indexing set. Choose K such that U is one of our
> > K_i's and denote the collection without U as K_j; that
> > is X = \/ K_i = U \/ (\/ K_j). Now since X is
> > connected U /\ (\/ K_j) must not be empty, for
> > otherwise it would be a separation. Thus
> > cl(U) /\ (\/ K_j) is non-empty and contains an
> > element of X - U.
> >
> > Maybe? I'm not 100% on this one, but that's what I came up with.
>
> A friend of mine just pointed out that the lemma in
> question asks about the closure of *some component* of U.
>
> I completely neglected that fact, and to be quite
> honest, I'm not sure what is meant by "component".
>
cl U is essential. A component of U is a maximal connected subset.
For a space, the component C(a) = \/{ U | a in connected U }
and the quasi-component, quasi-C(a) = /\{ U | a in clopen U }

When the space is compact Hausdorff
C(a) = quasi-C(a)
In general C(a) subset quasi-C(a)

Components of a subset are the components of the subspace.

The components of a space constitute a partition of closed connected sets.

> What is meant by "component" and does the approach still seem logical?
>
Think connected component and no, upon cursory reading, yicks.
.

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