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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Onto [0,1]
Posted: Apr 29, 2013 10:16 PM
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On Thu, 25 Apr 2013, Butch Malahide wrote:
> On Apr 25, 2:22 am, William Elliot <> 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.)

Indeed, the salient features
the intersection of a countable nest C, of not empty compact sets
with inf{ diam K | K in C } = 0, imples /\C is a singleton

and repleatedly decomposing a set into countably many closed sets

suffice to show N^N continuously maps onto the complete metric space.

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