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bill
science forum beginner
Joined: 09 Jul 2006
Posts: 12
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 Posted: Sat Jul 15, 2006 7:44 am Post subject:
continuity&metric spaces
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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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William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
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| Posted: Sat Jul 15, 2006 9:12 am Post subject:
Re: continuity&metric spaces
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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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José Carlos Santos
science forum Guru
Joined: 25 Mar 2005
Posts: 1111
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| Posted: Sat Jul 15, 2006 9:46 am Post subject:
Re: continuity&metric spaces
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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?
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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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