Every non-empty proper subset of Real Numbers is either open or closed. true/false

  • 2
    $\begingroup$ Hi! Welcome to math.stackexchange. Note that although we are a question and answer site, it is considered polite to phrase your question as a request rather than simply stating the problem verbatim. It would also be useful for members trying to help you to know what you have tried so far. $\endgroup$ – tharris Jun 5 '13 at 9:02
  • $\begingroup$ Here are some familiar subsets of $\Bbb R$: $[0,1]$, $(0,1)$, $[0,1)$, $\Bbb Q$. Can you tell me which of them are open? Closed? $\endgroup$ – Brian M. Scott Jun 5 '13 at 9:05
  • $\begingroup$ i m sorry, about the way i posted my question. i will keep in mind the next time for the questions: [0,1] closed (0,1)open [0,1) left-closed and right-open $\endgroup$ – Haffi Jun 5 '13 at 9:13

It is false.

For example, $[0,1)$ is not open, since $0 \in [0,1)$ has not a nbhd $U$ such that $0 \in U\subset [0,1)$. Also, $[0,1)$ is not closed, since the point 1 is accumulation of $[0,1)$, however 1 is not in $[0,1)$.


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