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metric spaces
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bill1158
science forum beginner


Joined: 09 Jul 2006
Posts: 12

PostPosted: Sat Jul 15, 2006 7:47 am    Post subject: metric spaces Reply with quote

(X, d) is an arbitrary metric space, and Y = R with d2.
f : X => R is continuous on X.
Can we show that for every c in R, the set {x in X : f(x) > c} is an
open set in X?
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Simple
science forum beginner


Joined: 15 Jul 2006
Posts: 3

PostPosted: Sat Jul 15, 2006 8:22 am    Post subject: Re: metric spaces Reply with quote

why here is a Y?

"bill" <bilgiaslankurt@gmail.com>
??????:1152949625.623276.271800@i42g2000cwa.googlegroups.com...
Quote:
(X, d) is an arbitrary metric space, and Y = R with d2.
f : X => R is continuous on X.
Can we show that for every c in R, the set {x in X : f(x) > c} is an
open set in X?
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Sat Jul 15, 2006 9:17 am    Post subject: Re: metric spaces Reply with quote

On Sat, 15 Jul 2006, bill wrote:

Quote:
(X, d) is an arbitrary metric space, and Y = R with d2.

What you mean
with d2?

Quote:
f : X => R is continuous on X.

Typo, X -> R. I'll consider R to the the reals with the usual
open interval topology.

Quote:
Can we show that for every c in R, the set {x in X : f(x) > c} is an
open set in X?

Yes. (c,oo) is an open set and since f is continuous

{ x | c < f(x) } = f^-1((0,oo))
is open.
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Sat Jul 15, 2006 10:02 am    Post subject: Re: metric spaces Reply with quote

On Sat, 15 Jul 2006, William Elliot wrote:

Typo *** corrected at end of this post.

Quote:
On Sat, 15 Jul 2006, bill wrote:

(X, d) is an arbitrary metric space, and Y = R with d2.

What you mean
with d2?

f : X => R is continuous on X.

Typo, X -> R. I'll consider R to the the reals with the usual
open interval topology.

Can we show that for every c in R, the set {x in X : f(x) > c} is an
open set in X?

Yes. (c,oo) is an open set and since f is continuous
{ x | c < f(x) } = f^-1((0,oo))

*** { x | c < f(x) } = f^-1((c,oo))

Quote:
is open.
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bill1158
science forum beginner


Joined: 09 Jul 2006
Posts: 12

PostPosted: Sun Jul 16, 2006 8:40 am    Post subject: Re: metric spaces Reply with quote

What if the set was a closed set in X where {x in X : f(x) greater than
or equal to c} is a closed set in X.?
What changes then for us to show the continuity of X?

Many thanks!
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Sun Jul 16, 2006 9:28 am    Post subject: Re: metric spaces Reply with quote

On Sun, 16 Jul 2006, bill wrote:

Quote:
What if the set was a closed set in X where {x in X : f(x) greater than
or equal to c} is a closed set in X.?
What changes then for us to show the continuity of X?

Don't know as the problem statement has been removed.

Ask your question again with context included.

-- To Google and MathForum users:
Reply only if adequate context is included _within_ the reply.
Otherwise all contexts are removed from my view,
the flow of thought disrupted and chaos reigns.
http://oakroadsystems.com/genl/unice.htm#quote

In particular for Google users:

Instead of simply hitting the prominent "Reply" link, which doesn't
include a copy of the post to which one is replying, click the "Show
Options" link (toward the top of an item in the thread), which causes
a shaded area of links to appear next to the top of the item, including
"Reply" (first) that does introduce a copy of the previous text (offset
by > signs in the usual fashion).
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bill1158
science forum beginner


Joined: 09 Jul 2006
Posts: 12

PostPosted: Sun Jul 16, 2006 2:34 pm    Post subject: Re: metric spaces Reply with quote

My question now is:
Let (X, d) be an arbitrary metric space, X -> R
is continuous on X. Is the set {x in X : f(x) greater than or equal to
c} closed set in X. for every c in R?
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Josť Carlos Santos
science forum Guru


Joined: 25 Mar 2005
Posts: 1111

PostPosted: Sun Jul 16, 2006 2:55 pm    Post subject: Re: metric spaces Reply with quote

On 16-07-2006 15:34, bill wrote:

Quote:
My question now is:
Let (X, d) be an arbitrary metric space, X -> R
is continuous on X.

Just say that the function is continuous. Adding "on X" is superfluous
and may give rise to some confusion.

Quote:
Is the set {x in X : f(x) greater than or equal to
c} closed set in X. for every c in R?

Yes, since its complement is an open set.

Best regards,

Jose Carlos Santos
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Mon Jul 17, 2006 2:31 am    Post subject: Re: metric spaces Reply with quote

On Sun, 16 Jul 2006, bill wrote:

Quote:
Let (X, d) be an arbitrary metric space, X -> R
is continuous on X. Is the set {x in X : f(x) greater than or equal to
c} closed set in X. for every c in R?

Yes. X may be any topological space.


{ x | f(x) >= c } = f^-1([c,oo])

is the continuous inverse image of a closed set, hence closed.
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