Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

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


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

Topic: Onto [0,1]
Replies: 40   Last Post: Apr 29, 2013 10:16 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 29, 2013 3:25 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Apr 29, 3:24 am, William Elliot <ma...@panix.com> wrote:
> On Sat, 27 Apr 2013, Butch Malahide wrote:
> > On Apr 27, 3:04 am, William Elliot <ma...@panix.com> wrote:
> > > > Correctly: every *nonempty* compact metric space is a continuous
> > > > image of the Cantor set. (Likewise, every nonempty separable
> > > > complete metric space is a continuous image of the space of
> > > > irrational numbers.)

>
> > > > Let Y be a nonempty compact metric space. Then, for some natural
> > > > n_1, Y is the union of n_1 nonempty closed sets of diameter < 1.
> > > > Next, for some natural n_2, each of those n_1 sets is the union of
> > > > n_2 (not necessarily distinct) nonempty closed sets of diameter <
> > > > 1/2. Next, for some natural n_3, each of the previously chosen
> > > > n_1*n_2 sets is the union of n_3 nonempty closed sets of diameter <
> > > > 1/3. And so on.

>
> > > Cover Y with { B(y,1) | y in Y }.  Thus Y is the union of n_1
> > > sets of the form cl B(y,1) with not empty interior.

>
> > Yes, obviously. [However, I specified sets of *diameter* *less* than
> > 1, so change B(y,1) to B(y,1/3).]

>
> It makes no difference if the diameter of sets is < 1/n or <= 1/n.
> What's important is the limit of the diameters is zero.
>
>
>
>
>

> > NOTATION: As usual [n] = {1, 2, 3, ..., n} for natural n. Y is the
> > union of nonempty closed sets Y_x (x in [n_1]) of diameter < 1. If x_1
> > in [n_1], x_2 in [n_2], ..., x_{k-1} in [n_{k-1}], then Y_{x_1,
> > x_2, ..., x_{k-1}} is the union of nonempty closed sets Y_{x_1,
> > x_2, ..., x_{k-1}, x_k} (x_k in [n_k]) of diameter < 1/k.

>
> > > > Use these covengs in the obvious way to define a continuous
> > > > surjection from the infinite product space X = D(n_1)xD(n_2)x. . . to
> > > > Y, where D(n) is a discrete space of cardinality n.

>
> > > Nothing obvious at all about it. f:X -> Y, x -> to the unique element of a
> > > nest of compact sets as constructed above.  It's a notational nightmare.

>
> > > Since such a nest can be constructed for each y in Y, f is surjective.
>
> > > Pointwise continuity of f, though seemingly possible, isn't apparent for
> > > the mess of details.

> > Here is the "mess of details":
>
> > Suppose x = (x_1, x_2, ...) in X, f(x) = y in Y. Let epsilon > 0 be
> > given. Choose n so that 1/n < epsilon. Let U = {u in X: u_i = x_i for
> > all i <= n}. Then U is a neighborhood of x in X, and
> > f(U) subset Y_{u_1, ..., u_n} = Y_{x_1, ..., x_n} subset B(y, 1/n)
> > subset B(y, epsilon),
> > Q.E.D.

> > > > Finally, observe that X is a continuous image (in fact a homeomorph
> > > > but we don't need that) of the Cantor set.

>
> > Actually, the construction as given allows the possibility that X is
> > finite. If we want to ensure that X is homeomorphic to the Cantor set
> > C, we can do that by requiring n_i > 1. But there is no need for that.

>
> Yet it's simpler and more direct than going through C^N.


It's (very slightly) simpler if you don't count the work needed to
prove the topological characterization of C, which is harder than what
we're proving here. The reader who does not already know that
characterization will probably not appreciate your invoking such a
difficult result to save an easy two-line argument, and would probably
prefer a more self-contained approach.




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

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.