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?
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.