|
| Science Forum Index » Mathematics Forum » Subbases of a Topology |
|
Page 1 of 1 |
|
| Author |
Message |
| Jose Capco |
 Posted: Sun Jul 17, 2005 3:32 am |
|
|
|
Guest
|
Dear NG,
Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G? I think this is not true, but I just want to verify it.
Thanks in advance
Sincerely,
Jose Capco |
|
|
| Back to top |
|
|
|
| Stephen J. Herschkorn |
| Posted: Sun Jul 17, 2005 3:32 am |
|
|
|
Guest
|
Jose Capco wrote:
[quote:7852e3eda0]Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G?
[/quote:7852e3eda0]
No. The collection of intervals {(-infty, a): a in Q} U {(a,
+infty): a in Q} are a subbase for the usual topology on R.
--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor in Central New Jersey and Manhattan |
|
|
| Back to top |
|
|
|
| William Elliot |
| Posted: Sun Jul 17, 2005 3:32 am |
|
|
|
Guest
|
On Mon, 18 Jul 2005, Jose Capco wrote:
[quote:852f0645b0]Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G? I think this is not true, but I just want to verify it.
[/quote:852f0645b0]
No. { (-oo,r), (r,oo) | r in R } is a subbase for R
and (0,1) is open set. |
|
|
| Back to top |
|
|
|
| William Elliot |
| Posted: Sun Jul 17, 2005 3:32 am |
|
|
|
Guest
|
On Mon, 18 Jul 2005, Jose Capco wrote:
[quote:bcb1d9392c]On Mon, 18 Jul 2005 08:48:35 +0200, Jose Capco <nospam@nospam.com> wrote:
Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G? I think this is not true, but I just want to verify it.
Now had S been a base, this would be true wouldn't it?
Almost. For it to be true,[/quote:bcb1d9392c]
nulset would have to be included in the base. |
|
|
| Back to top |
|
|
|
| quasi |
| Posted: Sun Jul 17, 2005 3:32 am |
|
|
|
Guest
|
On Mon, 18 Jul 2005 08:48:35 +0200, Jose Capco <nospam@nospam.com>
wrote:
[quote:c9259c1909]Dear NG,
Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G? I think this is not true, but I just want to verify it.
Thanks in advance
Sincerely,
Jose Capco
[/quote:c9259c1909]
Your intuition seems right here. After all, a subbase can use finite
intersections as well as arbitrary unions and so can be a lot smaller
than a basis.
Try the the standard topolgy on the reals. There is a natural basis
and a natural subbasis which is much smaller. Find them, and it will
answer your question. You can also try finite spaces (try a 3 point
space), but I think the reals better clarifies the concept.
-- quasi |
|
|
| Back to top |
|
|
|
| Jose Capco |
| Posted: Sun Jul 17, 2005 3:32 am |
|
|
|
Guest
|
On Mon, 18 Jul 2005 08:48:35 +0200, Jose Capco <nospam@nospam.com> wrote:
[quote:9e563cf0f5]Dear NG,
Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G? I think this is not true, but I just want to verify it.
Thanks in advance
Sincerely,
Jose Capco
[/quote:9e563cf0f5]
Now had S been a base, this would be true wouldn't it? |
|
|
| Back to top |
|
|
|
| Marc Olschok |
| Posted: Mon Jul 18, 2005 6:51 am |
|
|
|
Guest
|
Jose Capco <nospam@nospam.com> wrote:
[quote:a74ee5e9cf]Dear NG,
Given that I know a topology T of a set X and a subbasis S of T. Is it
true that for any open set G in T, there is an element in S that is a
subset of G? I think this is not true, but I just want to verify it.
[/quote:a74ee5e9cf]
Construct a counterexample with |X|=3, |T|=3 and |S|=2.
Marc |
|
|
| Back to top |
|
|
|
|
|
All times are GMT - 5 Hours
The time now is Sat Jul 31, 2010 4:38 pm
|
|
