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Math Help - topology - open sets vs neighborhoods

  1. #1
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    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?
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  2. #2
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    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.
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  3. #3
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    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".
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