# topology - open sets vs neighborhoods

• Jun 30th 2006, 11:43 AM
nweissma
topology - 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?
• Jun 30th 2006, 12:21 PM
Soltras
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, 04:08 PM
Copestone
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.

---
(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".