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Subbasis and Finer Topologies

  1. Dec 6, 2016 #1
    1. The problem statement, all variables and given/known data
    If two different topologies on a given space ##X## are given by a basis, then we have the following criteria for determining which topology is finer:

    Let ##\mathcal{B}## and ##\mathcal{B}'## be bases for the for topologies ##\tau## and ##\tau'##, respectively, on ##X##. Then the following are equivalent:

    (1) ##\tau \subseteq \tau'##

    (2) For each ##x \in X## and each basis element ##B \in \mathcal{B}##, there is a basis element ##B' \in \mathcal{B}'## such that ##x \in B' \subseteq B##.

    2. Relevant equations


    3. The attempt at a solution

    I am wondering, do we have a similar criteria if the two topologies are given by a subbasis ##\mathcal{S}## and ##\mathcal{S}'##? My motivation for asking this question is that I am trying to show that ##\{Y \cap S ~|~ S \in S \}## forms a subbasis for the subspace topology on ##Y \subseteq X##, and I thought that it might help me in proving this.
     
  2. jcsd
  3. Dec 6, 2016 #2

    fresh_42

    Staff: Mentor

    Do you mean by subbasis ##\mathcal{S}## a basis of the from ##(X,\mathcal{B})## induced topology of ##Y##?
     
  4. Dec 6, 2016 #3
    Here is the definition of subbasis with which I am working:

    A subbasis ##\mathcal{S}## for a topology on ##X## is a collection of subsets of ##X## whose union equals ##X##. The topology generated by the subbasis ##\mathcal{S}## is defined to be the collection ##\tau## of all unions of finite intersections of elements in ##\mathcal{S}##.
     
  5. Dec 6, 2016 #4

    fresh_42

    Staff: Mentor

    I can't see any problems. The topology generated by the sets of ##\mathcal{S}## is the least finest topology which contains these sets. Refinements thus should be possible to define analogously if everything is done with rigor. (But as topology is notoriously strange, I would be delighted to learn something new.)
     
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