1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show that B is not a topology on R

  1. Apr 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Let B be the family of subsets of [itex]\mathbb{R}[/itex] consisting of [itex]\mathbb{R}[/itex] and the subsets [n,a) := {[itex]r \in \mathbb{R} : n \leq r < a[/itex]} with [itex]n \in \mathbb{Z}[/itex], a [itex]\in \mathbb{R}[/itex] Show that B is not a topology on [itex]\mathbb{R}[/itex]


    2. Relevant equations



    3. The attempt at a solution
    If B were a topology then we would need:
    [itex]\emptyset[/itex]and [itex]\mathbb{R} \in[/itex] B (1), the arbitrary union of any opens in B to be in B (2) and any finite union of opens in B to be in B (3). Now the first two conditions (1), (2), seem to be valid so if B is not a topology on [itex]\mathbb{R}[/itex] then certainly condition (3) would have to fail. My question is, does condition (3) indeed fail and if it does, how can I show this?

    Thanks in advance
     
  2. jcsd
  3. Apr 11, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    What is the union of all of the [n,a)?
     
  4. Apr 11, 2013 #3
    If I am not mistaken the union of all of the [n,a) is [itex]\mathbb{R}[/itex]
     
  5. Apr 11, 2013 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes. What if you take a little bit less of the ##[n,a)##?? Can you form some half-open interval?
     
  6. Apr 11, 2013 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No, I don't think it's all of R. a or a+1 isn't in it. Oh, and your definition of topology is a little off. You want finite intersections to be in the topology. Specifying finite unions after you already said arbitrary union would be a little redundant.
     
    Last edited: Apr 11, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted