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!

Plz help me check my reasoning

  1. Apr 1, 2008 #1
    Is my reasoning correct?

    Original problem:
    In the finite completement topology on R(denoted by FCTR here), to what point or poionts does the sequense xn = 1/n converge?

    I firstly prove that R with FCT does not a Hausdorff.

    Let Tf be FCTR, x1, x2 are two arbitrary points of R, U1, U2 are their neighborhoods, respectively.

    Then R with Tf is not a Hausdorff, since:
    U1, U2 are open in R, U1, U2[tex]\in[/tex]Tf. R-U1 is finite, and R-U2 is finite. Then if U1[tex]\cap[/tex]U2=[tex]\phi[/tex] (which is necessary for a Hausdorff space), then R-(U1[tex]\cap[/tex]U2) will be R. Whereas R-(U1[tex]\cap[/tex]U2) = (R-U1)[tex]\cup[/tex](R-U2), which is finite, by definition, and impossible to be R. Hence U1 and U2 are not disjoint. Thus R with FCT does not a Hausdorff.

    Secondly I prove xn converge to every point of R.

    Since for every x in R with FCT, the neighborhood of x is the set Ux=R-{xn}. And for every Ux, all xn are in Ux, thus xn converge to x. For the arbitrary of x, xn converge to every point of R.

    Last edited: Apr 1, 2008
  2. jcsd
  3. Apr 1, 2008 #2
    Could someone kind enough to tell me, please?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?