compact space
stochastician
quasi
wrote:
>Is it possible to define a topology on N (the set of natual numbers) s.t. to make it a _compact_ space? Thanks! Obviously the trivial topology does not work.
Actually, the _trivial_ topology, where the open sets are just the
empty set and the full space, does work.
Perhaps you meant that the _discrete_ topology doesn't work.
quasi
quasi
Some ideas to try ...
(1) What if you take a topology on N which has only finitely many open
sets? Wouldn't it automatically be compact?
(2) How about cofinite sets? Consider covers of N using such sets.
quasi
stochastician
William Elliot
Is it possible to give N a compact Hausdorff topology?
Exercise.
Give an example of a countable, compact Hausdorff space.
José Carlos Santos
> Is it possible to define a topology on N (the set of natual numbers)
> s.t. to make it a _compact_ space?
Take the topology for which the open sets are:
1) All finite subsets of N of which 1 is not a member.
2) All subsets of N which contain a set of the form
{1, n, n + , n + 2, n + 3, ...}
for some natural _n_.
It is compact and metrisable.
Best regards,
Jose Carlos Santos
Philippe Gaucher
> On Sun, 13 Jan 2008, stochastician wrote:
>
>> Is it possible to define a topology on N (the set of natual numbers)
>> s.t. to make it a _compact_ space? Thanks! Obviously the trivial
>> topology does not work.
>>
> Is it possible to give N a compact Hausdorff topology?
Consider {0,1} with the discrete topology: it is compact
Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable
product) equipped with the product topology. It is compact Hausdorff
by Tychonoff's theorem. Any bijection between
{0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N.
pg.
José Carlos Santos
>>> Is it possible to define a topology on N (the set of natual numbers)
>>> s.t. to make it a _compact_ space? Thanks! Obviously the trivial
>>> topology does not work.
>>>
>> Is it possible to give N a compact Hausdorff topology?
>
> Consider {0,1} with the discrete topology: it is compact
> Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable
> product) equipped with the product topology. It is compact Hausdorff
> by Tychonoff's theorem. Any bijection between
> {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N.
Well, yes. But is is more natural to take, for instance
{0} U { 1/n | n natural }
with its natural topology. It is a compact Hausdorff space. No need to
use Tychonoff's theorem here.
brieucs
>> Is it possible to give N a compact Hausdorff topology?
> Consider {0,1} with the discrete topology: it is compact
> Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable
> product) equipped with the product topology. It is compact Hausdorff
> by Tychonoff's theorem. Any bijection between
> {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N.
do you mean that the product {0,1}x{0,1}x{0,1}x...
(countable product) is countable itself ?
quasi
Good point.
In fact, it's clearly _uncountable_.
quasi
Philippe Gaucher
> do you mean that the product {0,1}x{0,1}x{0,1}x...
> (countable product) is countable itself ?
Oopps sorry :-). I found a compact Hausdorff topology on R, not on N.
And {0,1}+{0,1}+{0,1}+... (disjoint sum) does not work either. It is
countable but not compact anymore.
pg.
Philippe Gaucher
> On 14-01-2008 10:35, Philippe Gaucher wrote:
>
>>>> Is it possible to define a topology on N (the set of natual numbers)
>>>> s.t. to make it a _compact_ space? Thanks! Obviously the trivial
>>>> topology does not work.
>>>>
>>> Is it possible to give N a compact Hausdorff topology?
>>
>> Consider {0,1} with the discrete topology: it is compact
>> Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable
>> product) equipped with the product topology. It is compact Hausdorff
>> by Tychonoff's theorem. Any bijection between
>> {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N.
>
> Well, yes.
Not well :-/. Here is another example which should work (I wanted to
give a conceptual example). Consider N with the discrete topology. It
is locally compact. I am a little bit confused between French and
English terminology: I mean Hausdorff and any point admits a compact
Hausdorff neighbourhood. Then consider the Alexandroff one-point
compactification. The space N u {oo} is still countable (no mistake
this time). Then using a bijection between N u {oo} and N, one obtains
a compact Hausdorff topology on N.
pg.
quasi
<p...@crepe.flambee> wrote:
>José Carlos Santos <jcsa...@fc.up.pt> writes:
>
>> On 14-01-2008 10:35, Philippe Gaucher wrote:
>>
>>>>> Is it possible to define a topology on N (the set of natual numbers)
>>>>> s.t. to make it a _compact_ space? Thanks! Obviously the trivial
>>>>> topology does not work.
>>>>>
>>>> Is it possible to give N a compact Hausdorff topology?
>
>Then consider the Alexandroff one-point compactification. The
>space N u {oo} is still countable (no mistake this time). Then
>using a bijection between N u {oo} and N, one obtains a
>compact Hausdorff topology on N.
That works perfectly,
quasi
G. A. Edgar
What is the relevance of that to your problem?
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
quasi
<ed...@math.ohio-state.edu.invalid> wrote:
>Is there a countable subset of the real line that is compact?
>What is the relevance of that to your problem?
Nice hint.
quasi
Zdislav V. Kovarik
Easier yet: the range of a convergent sequence with distinct points,
together with its limit, is a compact countable subset of the real line.
Make up your own bijection.
Cheers, ZVK(Slavek).