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 Post subject: Space of Continuous FunctionsPosted: Mon, 23 Jan 2012 15:54:46 UTC
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Reviewing my analysis here:

Let .
I want to prove that is not open. The most obvious approach for me is to show that the interior of is a proper subset, i.e., .
I'm just having trouble coming up with an example function.
I've been trying variants of the exponential function such as and it's not working.
I know I can write .

Any ideas?

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Space of Continuous FunctionsPosted: Mon, 23 Jan 2012 17:22:16 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12097
Location: Austin, TX
Justin wrote:
Reviewing my analysis here:

Let .
I want to prove that is not open. The most obvious approach for me is to show that the interior of is a proper subset, i.e., .
I'm just having trouble coming up with an example function.
I've been trying variants of the exponential function such as and it's not working.
I know I can write .

Any ideas?

What's the problem with the function ? It's not in any of those sets which form the interior, since there is no such lower bound...

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 Post subject: Re: Space of Continuous FunctionsPosted: Mon, 23 Jan 2012 18:51:19 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 990
What's the problem with the function ? It's not in any of those sets which form the interior, since there is no such lower bound...

Am I missing something? is not bounded on ...

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Space of Continuous FunctionsPosted: Mon, 23 Jan 2012 22:35:35 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12097
Location: Austin, TX
Justin wrote:
What's the problem with the function ? It's not in any of those sets which form the interior, since there is no such lower bound...

Am I missing something? is not bounded on ...

Oh right, so

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 Post subject: Re: Space of Continuous FunctionsPosted: Sun, 29 Jan 2012 20:04:04 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 990
Justin wrote:
What's the problem with the function ? It's not in any of those sets which form the interior, since there is no such lower bound...

Am I missing something? is not bounded on ...

Oh right, so

Yes, this turned out to work...I think I just needed someone to confirm my answer.

Thanks!

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Space of Continuous FunctionsPosted: Sun, 29 Jan 2012 22:27:56 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12097
Location: Austin, TX
Justin wrote:
Justin wrote:
What's the problem with the function ? It's not in any of those sets which form the interior, since there is no such lower bound...

Am I missing something? is not bounded on ...

Oh right, so

Yes, this turned out to work...I think I just needed someone to confirm my answer.

Thanks!

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

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