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

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 29, 2013 3:25 PM

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
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.

Date Subject Author
4/21/13 William Elliot
4/21/13 Butch Malahide
4/21/13 William Elliot
4/22/13 David C. Ullrich
4/22/13 Butch Malahide
4/22/13 David C. Ullrich
4/21/13 Bacle H
4/21/13 William Elliot
4/21/13 Bacle H
4/21/13 Bacle H
4/21/13 Butch Malahide
4/21/13 Bacle H
4/25/13 William Elliot
4/25/13 Butch Malahide
4/27/13 William Elliot
4/27/13 Butch Malahide
4/27/13 Butch Malahide
4/29/13 Butch Malahide
4/29/13 William Elliot
4/25/13 William Elliot
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 David C. Ullrich
4/21/13
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 Tanu R.
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/26/13 Butch Malahide
4/26/13 quasi