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José Carlos Santos science forum Guru
Joined: 25 Mar 2005
Posts: 1111

Posted: Sat Jul 15, 2006 9:46 am Post subject:
Re: continuity&metric spaces



bill wrote:
Quote:  Let X and Y be two metric spaces, and let the function f : X >Y
have the following property: for all sets E is a subset of X, we have
f(E closure) is a subset of (f(E)) closure.
Is f continous on X?

Yes.
Let _x_ be an element of X. If _f_ were not continuous at _x_, then
there would be a sequence (x_n)_n such that lim_n x_n = x and that
the distance from each f(x_n) to f(x) would be greater than a fixed
r > 0. Put E = { x_n  n natural }. Then f(cl(E)) would contain f(x),
but cl(f(E)) wouldn't. Therefore, f(cl(E)) would not be a subset of
cl(f(E)).
Best regards,
Jose Carlos Santos. 

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William Elliot science forum Guru
Joined: 24 Mar 2005
Posts: 1906

Posted: Sat Jul 15, 2006 9:12 am Post subject:
Re: continuity&metric spaces



On Sat, 15 Jul 2006, bill wrote:
Quote:  Let X and Y be two metric spaces, and let the function f : X >Y
have the following property: for all sets E is a subset of X, we have
f(E closure) is a subset of (f(E)) closure.
Is f continous on X? How can we show?
It is known for all topological spaces that f is continuous iff 
for all A, cl f^1(A) subset f^1(cl A)
Let E = f^1(A). Thus from the property
cl f^1(A) subset f^1f(cl f^1(A)
subset f^1(cl ff^1(A))
subset f^1(cl A)
showing f is continuous. cl E = E closure.
Conversely, if f is continuous, then for all E
f(cl E) subset cl f(E)
showing the property is equivalent to continuity,
not only just for metric spaces, but all spaces. 

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bill1158 science forum beginner
Joined: 09 Jul 2006
Posts: 12

Posted: Sat Jul 15, 2006 7:44 am Post subject:
continuity&metric spaces



Let X and Y be two metric spaces, and let the function f : X >Y
have the following property: for all sets E is a subset of X, we have
f(E closure) is a subset of (f(E)) closure.
Is f continous on X? How can we show? 

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