From wikipedia: "Now consider the space X which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set (0,1) is clopen, as is the set (2,3)."

why (0,1) is clopen?

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    $\begingroup$ Because its complement in $X$ is open $\endgroup$ – Omnomnomnom Jul 30 '16 at 12:03

The set $(0,1)$ is open as a subset of $\mathbb{R}$ and therefore also open in the subspace topology induced on $(0,1) \cup (2,3)$. The same holds for $(2,3)$. Since the complement of $(0,1)$ in $(0,1) \cup (2,3)$ is $(2,3)$ and therewith open, $(0,1)$ is also closed in $(0,1) \cup (2,3)$. Hence it is clopen.

  • $\begingroup$ ok thanks. another question: if we have as X=R, then [0,1) is clopen? and why? every ball with 0 as center doesn't belong into [0,1). so it's not open. the complement of [0,1) is A=(-oo,0)U[1,+oo) . every ball of this complement with 1 as center doesn't belong into A so it's not open, so [0,1) is not closed set. am i correct? $\endgroup$ – asdfar Jul 30 '16 at 12:20
  • $\begingroup$ @AlfaBhtaBasilopoylos $[0,1)$ is not clopen ... or open ... or closed. Basically subsets come in $4$ varieties: strictly open, strictly closed, clopen, and not closed or open. $[0,1)$ is not closed or open. $\endgroup$ – user137731 Jul 30 '16 at 12:26
  • $\begingroup$ @AlfaBhtaBasilopoylos You showed that [0,1) is neither open nor closed. Clopen sets are both open AND closed. Quite a difference ;-) $\endgroup$ – Tim B. Jul 30 '16 at 12:58
  • $\begingroup$ Though $[0, 1)$ is often called "clopen", it means more that it's half-closed and half-open. It's not a topological clopen, merely a convenient shorthand for a class of intervals. $\endgroup$ – AJY Jul 30 '16 at 13:01
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    $\begingroup$ DO NOT use the word "clopen" for $[0,1)$ in $\mathbb R$. Almost everyone will be confused by that. No doubt someone once did this, but I disagree that it is done "often". $\endgroup$ – GEdgar Jul 30 '16 at 13:44

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