My Math Forum

My Math Forum (http://mymathforum.com/math-forums.php)
-   Topology (http://mymathforum.com/topology/)
-   -   Topology generated from subbasis (http://mymathforum.com/topology/14413-topology-generated-subbasis.html)

martexel September 4th, 2010 11:34 AM

Topology generated from subbasis
 
I'm teaching myself topology from Munkres' book, and I ran across this question in the exercises:
Quote:

Show that if A is a basis for a topology on X, then the topology generated by A equals the intersection of all topologies on X that contain A. Prove the same if A is a subbasis.
Proving it isn't the problem - I can do that - but what's troubling me is the implication; is it saying that the topology generated by a basis A is equal to the topology generated by A as a subbasis? Because it seems like you're trying to prove that both topologies equal the intersection of all topologies on X that contain A [and are consequently equal themselves]. If so, then what's the use of using a subbasis if it's just going to end up working identically to a basis?

September 4th, 2010 12:48 PM

Re: Topology generated from subbasis
 
Hello martexel.

I think the question is asking you to prove that if A is a basis for a topology on X, then the topology generated by A equals the intersection of all the topologies on X containing A, and if alternatively A is a subbasis for a topology on X, then the topology generated by A equals the intersection of all the topologies on X containing A.

Therefore, it is not implicit in the question that the two topologies so generated be equal, since A as a basis in the first case and as a subbasis in the second are in no way related. It may help to subscript the two differently. The crux of the matter is how we define "the topology generated by a basis" versus "the topology generated by a subbasis", as well as the difference in the definition of "basis" and "subbasis". (Keep in mind that a basis is automatically a subbasis, so a subbasis is "easier" to produce.) Look over these differences again and your worries will disappear.

Regards,
Ormkärr


All times are GMT -8. The time now is 02:00 AM.

Copyright © 2019 My Math Forum. All rights reserved.