Definition talk:Local Basis

The set $\mathcal B$ doesn't need to be formed of open sets. --Dan232 16:21, 23 August 2011 (CDT)

Argh! This is doing my head in! Steen & Seebach specifically state that the sets have to be open.

This is sort-of backed up by Wolfram (which is actually a bit thin on topology). Wikipedia appears to favour the definition you mention.

PlanetMath is more complicated: "a local basis is a family of neighborhoods such that every neighborhood of x contains a member of the family." But then you look up "neighborhood" and it says : "The meaning of the word neighborhood in topology is not well standardized. For most authors, a neighborhood of a point x in a topological space X is an open subset U of X which contains x." and then: "More generally, a neighborhood of any subset S of X is defined to be an open set of X containing S". PlanetMath's grammar is less than perfect and it's difficult to tell exactly how the statements are independent (i.e. that there's a full stop missing) or are dependent (and what's missing is an "and" or "but" etc).

Some sources say one thing, some say another. I can't find a source which uses one definition but also acknowledges the existence of the other definition.

I understand you seem to be very positive and are certain of your knowledge (<irony>wish I was clever enough to know I was in no doubt</irony>) but this is getting us deeper into muddy waters of so many different concepts with different definitions.

Either there is "one true way, and all unbelievers are heretics", or there are several different approaches, each of which gives rise to its own system of theorems. Until we can be sure of that, we can't begin to say "This is right, that is wrong."

In my favour, I have a book in front of me in which the system of results is (apart from a few actual mistakes I've found) self-consistent. In your favour, you have a website nLab in front of you, which uses a completely different system of definitions for local properties, but I haven't had a chance to study it yet in any detail to be able to comment.

Come to think of it, are we even sure that a local basis (as defined here, all sets need to be open for it to apply) and a neighborhood base (as you use the term, with no requirement that the sets be open) are the same thing in the first place? --prime mover 16:51, 23 August 2011 (CDT)

I know that there are different definitions, I'm sorry if my statements are too strong (that's because English is not my native lenguage). Your definition here is too restrictive because, as you asked, all sets must be open for it to apply, and I guess that it is old-fashioned. I think that it is old-fashion because the newer definitions, are usually more general. I checked planetmath, and it uses the same definition as wikipedia. I don't want to tell you what to do and how to do it, so do whatever you think is best.

The function of a local basis is to recover all neighbourhoods from the ones forming the local basis, that is why it is not requiered to be open in some books.

Yes, local basis and neighbourhood basis are the same thing.--Dan232 17:16, 23 August 2011 (CDT)

I'm not sure PlanetMath does use the same definition as wikipedia. When I checked (see what I posted above), it says:

"a local basis is a family of neighborhoods such that every neighborhood of x contains a member of the family."

... and then on the page for "neighborhood":

"The meaning of the word neighborhood in topology is not well standardized. For most authors, a neighborhood of a point x in a topological space X is an open subset U of X which contains x." And it then goes on to say that if not specified on any page on PlanetMath, it uses the term "neighborhood" to mean "open neighborhood".

From which I understand it their definition to mean a family of open neighborhoods.

My problem is that if there are two definitions throughout the literature for the same thing, then it is going to be difficult to reconcile them. And the trouble with that is, I don't know what to do, at the moment. Let me think about it some more. --prime mover 00:43, 24 August 2011 (CDT)

Let me put in that the definition of Definition:Locally Compact Space does not make much sense at the moment, since compact, open sets tend to be... rare. --Lord_Farin (talk) 14:32, 6 October 2012 (UTC)

Not that rare. You'd be surprised. --prime mover (talk) 09:10, 3 November 2012 (UTC)

Just reviewing this thread, and indeed, I now get L_F's point: "compact, open sets tend to be... rare" because compact sets are by definition closed. :-)

The point is that the definition Definition:Locally Compact Space makes sense only if the definition Definition:Neighborhood Basis allows for the neighborhoods of the points to be closed. --prime mover (talk) 04:45, 21 January 2018 (EST)

As it stands, $\R$ is not locally compact according to the first definition (as open intervals aren't compact). But it is in the second, which contradicts Local Compactness in Hausdorff Space (and so I'm not surprised that that page hasn't got a proof yet). --Lord_Farin (talk) 09:44, 3 November 2012 (UTC)

Proposed Local Basis Changes

I have redone the pages for local basis. I have renamed the pages for Definition 1 and Definition 2 in the hope of clarifying the difference in the two definitions.

Definition:Local Basis

Definition:Local Basis/Local Basis for Open Sets

Definition:Local Basis/Neighborhood Basis of Open Sets

I have completed the theorem to show the equivalence of the two definitions.

Equivalence of Definitions of Local Basis

Local Basis for Open Sets Implies Neighborhood Basis of Open Sets

Neighborhood Basis of Open Sets Implies Local Basis for Open Sets

The equivalence of the definitions renders the theorem Local Basis is Neighborhood Basis unnecessary. The exisiting links to theorem Local Basis is Neighborhood Basis can be replaced with links to the definition Local Basis for Open Sets Implies Neighborhood Basis of Open Sets. Then the page Local Basis is Neighborhood Basis can be deleted.

I have created this theorem to help clarify the relationship between local basis and neighborhood basis

Local Basis Generated from Neighborhood Basis

The definition of a neighborhood basis is updated with the changes above.

Definition:Neighborhood Basis

Let me know if this is OK to go ahead and change —-Leigh.Samphier (talk) 07:14, 22 June 2019 (EDT)

Good job. I'll take a good look in due course, but I have immediate domestic commitments today. --prime mover (talk) 07:27, 22 June 2019 (EDT)

Looks fine to me. Please go ahead with your proposed changes. --prime mover (talk) 13:31, 22 June 2019 (EDT)