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!

First countable spaces and metric spaces.

  1. Jul 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Show That every metric space is first countable. Hence show that every SUBSET of a metric space is the intersection of a countable family of open sets.

    2. Relevant equations
    no equation

    3. The attempt at a solution
    its easy to show that it is first countable, because for every point in the space there is the set of rational open balls which are included in every other open set.
    but the second part of the question is confusing:
    how can every subset be an intersection of a countable family? we only know that at every point there is a countable family but. there maybe an uncountable number of point..

    Thanks for the help, i've been staring at the question for two hours..
    by the way this problem is from P. Szekeres chapter 10 problem 10.9
     
    Last edited: Jul 4, 2011
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: First countable spaces and metric spaces.
  1. Phase space (Replies: 0)

  2. Hausdorff Spaces (Replies: 0)

  3. Quotient space (Replies: 0)

Loading...