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My book defines disconnectedness in the following way: "If $M$ has a proper clopen subset $A,$ then $M$ is disconnected. Otherwise, $M$ is connected."

The book gives the following example. "The punctured interval $X = [a,b] \setminus \{c\}$ is disconnected, for $X = [a,c) \sqcup (c,b]$ is a separation of $X.$" I see that $[a,c),(c,b] \subset X,$ but is it not the case that $[a,c)$ and $(c,b]$ are not clopen (indeed, they are neither closed nor open)?

Maybe he is considering a space that is not $\mathbb{R}?$ The author goes on to say: "The closures of the two sets with respect to the metric space $X$ do not intersect, even though their colsures with respect to $\mathbb{R}$ do intersect."

Moreover, he follows that sentence with "Pay attention to this phenomenon, which is related to the Inheritance principle." What does he mean? How is it related to the inheritance principle?

Rafael Vergnaud
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  • They are open in the topology induced by $[a,b]$. Note that $[a,c)$ is an open subset of $[a,b]$, and thus open in $[a,c)\cup (c,b]$. Look up https://math.stackexchange.com/questions/1299036/what-is-induced-topology – Kolja Oct 20 '18 at 20:45
  • Yes, it is not "clopen" (I've always disliked that word). in R, it is "clopen" in X itself. – user247327 Oct 20 '18 at 20:46
  • So, if we were considering the space $\mathbb{R},$ the interval would be connected, but if we were considering the space $X,$ it would be? – Rafael Vergnaud Oct 20 '18 at 20:50
  • Must the subsets of the set $X$ under consideration only be clopen with respect to $X,$ even if we are considering $X$ in a space in which the subsets are not clopen, such as $\mathbb{R}?$ – Rafael Vergnaud Oct 20 '18 at 20:55

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When $(A,\tau)$ is a topological space and $Y\subset A$ then there is a subspace topology $\tau_Y=\{U\cap Y:U\in \tau\}$ on $X$. In your case $A$ and $Y$ are like $\mathbb{R}$ and $X$ respectively. Since $(a-1,c)$ is open in $\mathbb{R}$ then $(a-1,c)\cap X=[a,c)$ is open in $X$. Since $[a-1,c]$ is closed in $\mathbb{R}$ then $[a-1,c)]\cap X=[a,c)$ is closed in $X$. So $[a,c)$ is clopen in $X$. With almost same argument we can say $(c,b]$ is clopen in $X$. So it is obvious that their closure in $X$ themselves but in $\mathbb{R}$ intersects.

S.S.Danyal
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  • So, $X$ is only disconnected if the space we are considering is $X$ itself? If the space were $\mathbb{R},$ then the interval given in the example would be connected? – Rafael Vergnaud Oct 20 '18 at 21:00
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    @Rafael Vergnaud While talking on connectedness always we consider the space's topology. So there is no difference between to say "$X$ is disconnected in itself "and in "$X$ is disconnected in $\mathbb{R}$".We will consider always its own subspace topology. – S.S.Danyal Oct 20 '18 at 21:10
  • Ok. Thank you for the clarification, Danyal! – Rafael Vergnaud Oct 20 '18 at 21:17