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

Topic: Perfectly Uncountable
Replies: 27   Last Post: Jul 2, 2012 4:17 AM

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quasi

Posts: 9,080
Registered: 7/15/05
Re: Perfectly Uncountable
Posted: Jun 26, 2012 2:53 PM
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Frederick Williams wrote:
>
>quasi wrote:
>>
>> Frederick Williams wrote:
>> >
>> >William Elliot wrote:
>> >>
>> >> Finally, does C continuously map onto [0,1]?
>> >
>> >For every compact metric space X, there is a continuous
>> >f:C -> X.
>>
>> For any topological space X, there is a continuous function
>> f:C -> X (for example, any constant function).
>>
>> But the question asked about a _surjective_ continuous
>> function, not just a continuous function.
>
>Sorry.

No, it was my mistake, not yours.

Every compact metric space _is_ the continuous image of the
Cantor set. This is theorem 30.7 of Stephen Willard's text
"General Topology".

Quite a surprising result, I think.

quasi

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