a topologicalneighborhoodis clinically defined in terms ofopen sets- a distinction without a difference. Whatisthe difference?

consider: the separation axioms are defined itoopen sets; can they be defined itoneighborhoods? how?

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- June 30th 2006, 12:43 PMnweissmatopology - open sets vs neighborhoods
a topological

__neighborhood__is clinically defined in terms of__open sets__- a distinction without a difference. What*is*the difference?

consider: the separation axioms are defined ito__open sets__; can they be defined ito__neighborhoods__? how? - June 30th 2006, 01:21 PMSoltras
The difference is context.

A neighborhood*is*an open set, but when we use the term neighborhood, we usually are referring to the set of points closer than a certain radius around a specific point.

When we talk about open sets, we are talking about a set where every point has at least one neighborhood that is completely contained in the set (i.e. it doesn't necessarily have a metric "circular" shape like a neighborhood.). Also, some open sets can be the union of two separated open sets, but neighborhoods cannot be. - May 23rd 2007, 05:08 PMCopestone
Well, the definition for "neighborhood" may not be standard in topology (some authors define neighborhoods to be open) but very often you will see the following usage:

In a topological space X, a neighborhood of a point p is a set N such that there is an open set U contained in N and p is in U. That is, N is a set that contains p in its interior.

Another way to say it:

N is a neighborhood of p if and only if p is an interior point of N.

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(1) It makes it convenient to talk about objects such as closed and compact neighborhoods.

(2) The neighborhood system at a given point p is actually a "filter".