In one of my exercises, I had a statement of the form $\forall x \in (c,b]: \varphi(x)$. However, it was possible that $c$ could take on the value of $b$...in which case, $(c,b]=(b,b]$.
Is it safe to say that $(b,b] = b$? Or do I need to reword my initial statement to $\big(\forall x \in (c,b): \varphi (x) \big) \land \varphi(b)$
In the event that it matters, this exercise takes place within the context of $\mathbb R$.
Edit:
As the comments suggest...
$S_{\mathbb R}=(b, b] \iff S_{\mathbb R}=\{x \in \mathbb R: x \gt b \text{ and } x \leq b\}$. No such $x$ satisfies this. So $S_{\mathbb R} = \emptyset$.