1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Close of a Singleton in R1

  1. Mar 10, 2013 #1
    Let me add some clarification here, since I realized that my notation might be confusing. By [itex]\left\{x\right\}'[/itex] below, I mean the set of all limit points of [itex]\left\{x\right\}[/itex]. Hope this helps anyone trying to figure out what I mean, thanks!

    The title says R1, but it's actually just a metric space [itex]X[/itex], my apologies.

    1. The problem statement, all variables and given/known data

    Let [itex]X[/itex] be a metric space and let [itex]x\in{X}[/itex] be any point. Prove that the set [itex]\left\{x\right\}[/itex] is closed in [itex]X[/itex]

    2. Relevant equations


    3. The attempt at a solution

    Ok, so I realize I can prove this by showing that the complement is open (simple). However, I have a question about the validity of an alternative approach:

    It is clear that [itex]\left\{x\right\}[/itex] has no limit points, thus [itex]\left\{x\right\}' = [/itex]empty set. Now, the empty set is a subset of every set, thus [itex]\left\{x\right\}'\subset{\left\{x\right\}}[/itex] and so [itex]\left\{x\right\}[/itex] contains all of its limit points. Then [itex]\left\{x\right\}[/itex] is closed.

    Thoughts? Does this work? If not, what am I missing? Thanks!
    Last edited: Mar 10, 2013
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted