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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'locally compact'
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Definition of neighborhood by AxelBoldt
Correction id: 666
Filed on: 2002-06-06 22:22:36
Status: Rejected on 2002-06-20 02:01:15
Type: Erratum
Correction text:
| The article uses the term "neighborhood" in a somewhat non-standard manner; most authors define a neighborhood of x to be a set which contains an open set containing x. |
Comment from object owner djao:
After further consideration, I must say that while I appreciate the tip, I am going to have to reject this correction.
The most immediate reason for the rejection is that, even if you are right and neighborhood should mean any set containing a nonempty open set, it does not change the validity of this definition: indeed, the two resulting notions of locally compact are equivalent regardless of which version of neighborhood is used.
As to the actual issue of what "neighborhood" should mean, a quick survey turns up the following answers:
* Munkres "Topology" defines a neighborhood to be an open set, as I have it.
* Ahlfors "Complex Analysis" defines a neighborhood to be a set containing an open set, as you have it.
* Rudin "Principles of Mathematical Analysis" defines a neighborhood to be an open ball of radius r.
The returns are, at best, inconclusive. My gut feeling is that the middle road adopted by Munkres is the best choice. It is not overly restrictive like Rudin, but it isn't as broad as Ahlfors either. Also, the Munkres book is far and away the standard introductory text used in topology, so I think deference must be given to it since the subject matter discussed in this entry is inherently topological in nature.
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