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Thread: What are the connectness and compactness of these sets?

  1. #1
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    What are the connectness and compactness of these sets?

    R with usual topology
    R\{0} .....
    Z with usual topology
    Z\{0} ....
    Q with usual topology
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    in the usual topology on these guys, compact means the set is closed and bounded. so i will leave you to wrestle with that

    as for connectedness
    Quote Originally Posted by szpengchao View Post
    R with usual topology
    R\{0} .....
    Note that $\displaystyle \mathbb{R} \backslash \{ 0 \} = (- \infty, 0) \cup (0, \infty)$

    Z with usual topology
    Z\{0} ....
    note that $\displaystyle \mathbb{Z} \backslash \{ 0 \} = \{ x \mid x \in \mathbb{Z},~x \in (-\infty, -1] \cup [1, \infty) \}$


    Q with usual topology
    can you express this set as two disjoint open sets?
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