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bill
science forum beginner
Joined: 09 Jul 2006
Posts: 12
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 Posted: Sat Jul 15, 2006 7:47 am Post subject:
metric spaces
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(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
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| Posted: Sat Jul 15, 2006 8:22 am Post subject:
Re: metric spaces
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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
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| Posted: Sat Jul 15, 2006 9:17 am Post subject:
Re: metric spaces
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On Sat, 15 Jul 2006, bill wrote:
| Quote: | (X, d) is an arbitrary metric space, and Y = R with d2.
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What you mean
with d2?
| Quote: | f : X => R is continuous on X.
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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
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| Posted: Sat Jul 15, 2006 10:02 am Post subject:
Re: metric spaces
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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))
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*** { x | c < f(x) } = f^-1((c,oo))
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bill
science forum beginner
Joined: 09 Jul 2006
Posts: 12
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| Posted: Sun Jul 16, 2006 8:40 am Post subject:
Re: metric spaces
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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
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| Posted: Sun Jul 16, 2006 9:28 am Post subject:
Re: metric spaces
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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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bill
science forum beginner
Joined: 09 Jul 2006
Posts: 12
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| Posted: Sun Jul 16, 2006 2:34 pm Post subject:
Re: metric spaces
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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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José Carlos Santos
science forum Guru
Joined: 25 Mar 2005
Posts: 1111
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| Posted: Sun Jul 16, 2006 2:55 pm Post subject:
Re: metric spaces
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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.
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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?
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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
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| Posted: Mon Jul 17, 2006 2:31 am Post subject:
Re: metric spaces
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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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