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Questions about the definition of open sets

  1. Nov 19, 2013 #1
    I am currently reading Munkres' book on topology, in it he defines an open sets as follows:
    "If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T."

    Firstly, are the open sets a property of the set X or the topological space (X,T)? Because if the open sets are all the things in the collection T of a particular topological space, different topological spaces on the same set would have different open sets. Making the whole notion of open sets of a set ill defined.

    Which brings me to my next question: are the open sets the sets in all possible collections T for a set X (in which case defining the open sets of a set makes sense), or are the open sets of a topological space the sets in the collection T of that topological space? In the second case I guess saying the open sets of X would have the fact that the open sets are a property of (X,T) implied. Since also the collection T itself is implied.
     
    Last edited: Nov 19, 2013
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  3. Nov 19, 2013 #2

    pasmith

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    Openness is a property of topological spaces. A set [itex]U \subset X[/itex] is open in [itex](X,\mathcal{T})[/itex] if and only if [itex]U \in \mathcal{T}[/itex].
     
  4. Nov 19, 2013 #3
    To elaborate:

    Given a set [itex]X[/itex], there's no innate meaning to an "open subset of [itex]X[/itex]". When somebody refers to, for example, an open subset of [itex]\mathbb R[/itex], what they mean is an element of [itex]\mathcal T_{\mathbb R}[/itex], which is just the usual (i.e. the one defined by absolute value) distance on [itex]\mathbb R[/itex].
     
  5. Nov 19, 2013 #4
    It's not ill defined, there is just more than one possible topology on a set X.
     
  6. Nov 19, 2013 #5
    Its the second case. Since in general there are many topological spaces for one set, [itex]X[/itex], so the open sets of a topological space of [itex]X[/itex] will depend on which topological space of [itex]X[/itex] you are working with.
     
  7. Nov 20, 2013 #6
    Open sets are completely determined by the topology on the set. Consider the indiscrete topology; only the empty set and the set itself are open (clopen). Why? Because that collection of the empty set and the point set itself meet all the requirements of being a topology, and we call the members of a topology "open sets."


    If you've taken linear algebra - it's exactly how "vectors" are just members of a set we can call a "vector space." In topology, "open sets" are just member of a collection of sets that we can call a "topology."

    If you can construct of a family of subsets of X such that it includes X, includes the empty set, and includes any union and finite intersection of other members of the family, you have a topology, and its members are called open sets.
     
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