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Math Help - basic topology trouble

  1. #1
    Senior Member abhishekkgp's Avatar
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    basic topology trouble

    I am reading 'Basic topology- M. A. Armstrong' and in Chapter 2 my book defines:

    Open set: Let X be a topological space then O \subset X is an open set if its a neighborhood of each of its points.
    Neighborhood: N \subset X is a neighborhood of x \in X if we can find an open set O such that x \in O \subset N.

    To define open sets we needed neighborhoods and to define neighborhoods we needed open sets. This seems ambiguous to me. Help needed on this.
    Also if you know a good book on Topology for Beginners then please tell me. I have read 'A first course in Real Analysis- Sterling K. Berberian'.
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    A Plied Mathematician
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    Re: basic topology trouble

    That does seem a bit circular. Any chance you could scan in and post the pages that have those two definitions?

    Off-hand, I would say that, typically, open sets are defined as the members of a topological space as follows: a topological space is a set, X, together with a collection, T, of subsets of X, called "open" sets, which satisfy the following rules:

    T1. The set X itself is "open"
    T2. The empty set is "open"
    T3. Arbitrary unions of "open" sets are "open"
    T4. Finite intersections of "open" sets are "open".

    - Martin D. Crossley, Essential Topology, p. 15.

    You can define open sets in \mathbb{R} and higher-dimensional spaces (really, any metric space) by looking at open balls, which is a more fundamental concept than open sets. In that setting, I would define a neighborhood, N, of x as a set such that there exists an open ball containing x that is itself contained in N. The concept of open balls does not depend on open sets, but it does need a metric.

    The Crossley book, incidentally, is one I would recommend. Explanations seem very straight-forward to me.
    Last edited by Ackbeet; August 5th 2011 at 06:30 AM.
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    Re: basic topology trouble

    Quote Originally Posted by abhishekkgp View Post
    Open set: Let X be a topological space then O \subset X is an open set if its a neighborhood of each of its points.
    It can't be the definition of an open set : the term "topological space" implies the notion of open sets ! Indeed, if (X,T) is a topological space, as Ackbeet said, an open set is the name of the element of T.

    So your quote is just a property of an open set, not a definition.
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  4. #4
    Senior Member abhishekkgp's Avatar
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    Re: basic topology trouble

    Quote Originally Posted by Ackbeet View Post
    That does seem a bit circular. Any chance you could scan in and post the pages that have those two definitions?

    Off-hand, I would say that, typically, open sets are defined as the members of a topological space as follows: a topological space is a set, X, together with a collection, T, of subsets of X, called "open" sets, which satisfy the following rules:

    T1. The set X itself is "open"
    T2. The empty set is "open"
    T3. Arbitrary unions of "open" sets are "open"
    T4. Finite intersections of "open" sets are "open".

    - Martin D. Crossley, Essential Topology, p. 15.

    You can define open sets in \mathbb{R} and higher-dimensional spaces (really, any metric space) by looking at open balls, which is a more fundamental concept that open sets. In that setting, I would define a neighborhood, N, of x as a set such that there exists an open ball containing x that is itself contained in N. The concept of open balls does not depend on open sets, but it does need a metric.

    The Crossley book, incidentally, is one I would recommend. Explanations seem very straight-forward to me.
    thank you Ackbeet. I will check out the book you recommended.
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    A Plied Mathematician
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    Re: basic topology trouble

    Quote Originally Posted by abhishekkgp View Post
    thank you Ackbeet. I will check out the book you recommended.
    You're welcome!
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  6. #6
    Senior Member Tinyboss's Avatar
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    Re: basic topology trouble

    The standard book recommendation (and it's the standard for good reason) is Topology by Munkres.
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