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?
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?
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.
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".