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Ali Taghavi
science forum addict
Joined: 14 May 2005
Posts: 73
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 Posted: Sun Feb 20, 2005 1:15 am Post subject:
second countable metric Spaces
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Let X be a second countable metric spaces(X has a countable dense
subset)
can we assigne a unique topology to X(The Topology Of its countable
dense subset)??namely if Y and Z be two countable dense subsets is Y~Z
As a sample:are the following two topological spaces homeomorph?:
rational number and Algebraic irrational nummber(both are countable
dense subsets of R)
thanks
Ali Taghavi
Iran |
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William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
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| Posted: Sun Feb 20, 2005 1:45 pm Post subject:
Re: second countable metric Spaces
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| Quote: | From: Ali Taghavi <taghavi@ipm.ir
Newsgroups: sci.math.research
Subject: second countable metric Spaces
Let X be a second countable metric spaces(X has a countable dense
subset) can we assigne a unique topology to X(The Topology Of its
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Is this last X the same or different than the X's in the first line?
| Quote: | countable dense subset)??namely if Y and Z be two countable dense
subsets is Y~Z
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Because X is 2nd countable metric, it has at most countably many
open singletons S = { x | {x} open }. Let Y be countable dense
subset. Then S subset Y; Y\S homeomorphic Q by theorem below.
When S is finite Y = disjoint sum Y\S + S homeomorphic
disjoint sum Q + F where F is finite discrete space of size |S|.
The same also for any other countable dense set of X.
When S infinite, X may have non-homeomorphic countable dense subsets.
For example X = R/\(-oo,0] \/ 1/N where 1/N = { 1/n | n in N }
has countable dense subset Q/\(-oo,0] \/ 1/N and Q/\(-oo,0) \/ 1/N.
Or X = 1/N \/ {0} with countable dense subsets X and 1/N.
Thus if X is 2nd countable regular T0 space with at most finite
many open singletons or if X is countable discrete space,
countable dense subsets of X are homeomorphic.
| Quote: | As a sample:are the following two topological spaces homeomorph?:
rational number and Algebraic irrational nummber(both are countable
dense subsets of R)
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Yes, by a Theorem of Sierpinski (circa 1920)
A space S is homeomorphic to the rationals if it is:
countable, 2nd countable, regular T0, free of isolated points.
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</x-flowed> |
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Ali Enayat
science forum beginner
Joined: 16 Feb 2005
Posts: 8
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| Posted: Mon Feb 21, 2005 12:45 pm Post subject:
Re: second countable metric Spaces
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On 19 Feb 2005 21:15:00 -0500, Ali Taghavi wrote:
| Quote: | Let X be a second countable metric spaces(X has a countable dense
subset)
can we assigne a unique topology to X(The Topology Of its countable
dense subset)??namely if Y and Z be two countable dense subsets is
Y~Z
As a sample:are the following two topological spaces homeomorph?:
rational number and Algebraic irrational nummber(both are countable
dense subsets of R)
|
The answer is "yes", and follows from a classical theorem of
Sierpinski asserting that all countable metric spaces with no isolated
points are homeomorphic to the rationals.
Regards,
Ali Enayat |
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