Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Subbasis vs. basis

  1. Feb 11, 2009 #1
    This is probably a stupid question.. but

    Can someone tell me the difference between subbasis and basis.. in topology?? I know the definitions...

    So Subbasis is defined to be the collection T of all unions of finite intersections of elements of S (subbasis)

    sooo... S is pretty much a topology on X which is a collection of subsets of X whose union equals X.

    Basis, however... is
    If X is a set, basis on X is a collection B of subsets of X (basis elements) s.t.
    1. for each x [tex]\in[/tex] X, there is at least one basis element B containing x.
    2. If x belongs to the intersection of two basis elements B1 and B2, then there is a basis element B3 containing x such that B3[tex]\subset[/tex] B1[tex]\cap[/tex]B2.

    Right? So pretty much... A subset U of X is said to be open in X if for each x [tex]\in[/tex] U, there is a basis element B [tex]\in[/tex] [tex]B[/tex] such that x [tex]\in[/tex] B and B [tex]\subset[/tex] U.

    But I'm still not understanding this quite... so well..

    Can someone explain this to me??


    Thank You!
     
  2. jcsd
  3. Feb 12, 2009 #2
    The topology generated by a subbasis is the topology generated by the basis of all finite intersections of subbasis elements. It seems like you just need an example, though, so here's one.

    A basis for the standard topology of R is the collection of all open intervals. A subbasis for R is the collection [tex]\{(a,\infty):a\in \mathbb{R}\} \cup \{(-\infty,b):b\in \mathbb{R}\}[/tex]. The reason that this is a subbasis for R is because finite intersections of elements in this set are precisely the basis elements of R. For instance, we have [tex](a,b) = (-\infty,b)\cap (a,\infty) [/tex]. Subbases are often important because they offer an easier way of expressing a topology. In the previous example, there are many fewer sets in the sub-basis than there are in the basis.
     
  4. Feb 12, 2009 #3
    I think there is a distiction that can help you avoid confusion.

    If you have a set [tex]X[/tex] and NO topology on it
    every [tex]S[/tex] subset of the set [tex]P(X)[/tex] of the parts of [tex]X[/tex]
    can be taken and you can generate e topology that has [tex]S[/tex] as subbasis.

    If you have a set [tex]X[/tex] and NO topology on it then
    if you have a set [tex]B \subset P(X)[/tex]
    that satisfies the 2 property 1. ans 2. you mentioned you can generate a topology on
    [tex]X[/tex] that has [tex]B[/tex] has basis.


    Now if you have a set [tex]X[/tex] and HAVE a topology [tex]T[/tex] on it
    a set [tex]S \subset P(X)[/tex] is a subbasis of [tex](X,T)[/tex] if and only if (by definition)
    for every open set [tex]A[/tex] and for every [tex]a \in A[/tex] exists
    [tex]S_1, \ldots, S_n \in S[/tex] such that
    [tex]a \in S_1 \cap \cdots S_n \subset A[/tex].


    Now if you have a set [tex]X[/tex] and HAVE a topology [tex]T[/tex] on it
    a set [tex]B \subset P(X)[/tex] is a basis of [tex](X,T)[/tex] if and only if (by definition)
    for every open set [tex]A[/tex] and for every [tex]a \in A[/tex] exists
    [tex]B_1 \in B[/tex] such that
    [tex]a \in B_1 \subset A[/tex].



    There is a slight difference to understand.
    Please read it carefully and think on it and you'll get the concept.
    Hope this helps.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Subbasis vs. basis
  1. Noncoordinate basis (Replies: 17)

  2. Basis ambiguity (Replies: 8)

  3. Basis topology (Replies: 3)

  4. Subbasis help (Replies: 0)

Loading...