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: Uncountable?
Replies: 9   Last Post: Apr 26, 2014 2:13 PM

Advanced Search

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

Posts: 12,067
Registered: 7/15/05
Re: Uncountable?
Posted: Apr 25, 2014 3:31 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Virgil wrote:
>William Elliot wrote:

>>Let C be a collection of open subsets of R that cover Q.
>>Can S = R - \/C ever be uncountable?

>It cannot even be non-empty!
>Every open cover of Q as a subset of R covers R.

That's not true.

As a simple example, let

A = (-oo,sqrt(2))

B = (sqrt(2),oo))

Then A \/ B covers Q but doesn't cover R.

If fact, there exist open covers of Q with uncountable complement.

For a construction, see my reply to William Elliot's first thread
on this question.


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.