The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Compactification
Replies: 15   Last Post: Mar 17, 2013 6:11 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Compactification
Posted: Dec 13, 2012 3:15 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Tue, 11 Dec 2012, Arturo Magidin wrote:
> On Tuesday, December 11, 2012 10:13:06 PM UTC-6, William Elliot wrote:

> > (h,Y is a (Hausdorff) compactification of X when h:X -> Y is an embedding,
> > Y is a compact (Hausdorff) space and h(X) is a dense subset of Y.
> >
> > Why the extra luggage of the embedding for the definition of
> > compactification? Why isn't the definition simply
> >
> > Y is a compactification of X when there's some
> > embedding h:X -> Y for which h(X) is a dense subset of Y?

> Compactification usually carry an (implicit) universal property: given
> any compact space Z and continuous function g:X-->Z, there exists a
> unique continuous G:Y-->Z such that g = Gh.

Is that true of all compactifications, that they have the universal
property? I wouldn't think so but haven't found a counter example.
Is there an example for the two point compactification [0,1] of R,
that shows it doesn't have the universal property?

> E.g. the Stone-Czech compactification is of this kind.

> This also provides universality and uniqueness, which makes it more
> useful in many circumstances.

Aren't two universal compactifications of the same space, homeomorphic?
How is uniqueness used? It didn't seem necessary for homeomorphism
of universal compactifications (of the space).

> > I see no advantage to the first definition. The second definition
> > has the advantage of being simpler and more intuitive. So why is
> > it that the first is used in preference to the second which I've
> > seen used only in regards to one point compactifications?

> Why this is the only one you've seen is probably an artifact of where
> you've looked. I've seen both definitions.

The two part definition seems more popular.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.