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Forum index » Science and Technology » Math
continuity&metric spaces
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Josť Carlos Santos
science forum Guru


Joined: 25 Mar 2005
Posts: 1111

PostPosted: Sat Jul 15, 2006 9:46 am    Post subject: Re: continuity&metric spaces Reply with quote

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.

Quote:
How can we show?

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

PostPosted: Sat Jul 15, 2006 9:12 am    Post subject: Re: continuity&metric spaces Reply with quote

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

PostPosted: Sat Jul 15, 2006 7:44 am    Post subject: continuity&metric spaces Reply with 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?
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