$\mathbb R_l$ is not connected.

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Solution 1

Any open set $[a,b)$ can be written as $[a,c) \cup [c,b)$ for $c$ between $a$ and $b$, so all open sets can be decomposed into non-empty, disjoint open sets.

Solution 2

The sets $[0,\infty)$ and $(-\infty,0)$ are both open in $\Bbb R_l$.

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Updated on August 01, 2022

Comments

  • Gauss
    Gauss over 1 year

    How to show $\mathbb R_l$ (lower limit topology on $\mathbb R$) is not connected?Means how any basis element of $\mathbb R_l$ can be written as the union of two separated sets?

    • Angina Seng
      Angina Seng over 6 years
      Can you please remind us what $\Bbb R_l$ is?
    • Admin
      Admin over 6 years
      @LordSharktheUnknown $\mathbb R_l$ is a well-known notation for $\mathbb R$ with the lower-limit topology (the topology generated by the basis $\{[a,b): a,b \in \mathbb R\}$).
    • Angina Seng
      Angina Seng over 6 years
      @OpenBall Could you then please remind us of a basis of $\Bbb R_l$?
    • Admin
      Admin over 6 years
      @LordSharktheUnknown sure! I just edited my comment.
  • Gauss
    Gauss over 6 years
    :i just read $\mathbb R_l$ is finer than $\mathbb R$ but not getting how to use this fact in the problem.thanks a lot!!