I come from an engineering background, so my knowledge of measure theory is very bad. Please pardon me for asking this silly question.
Let $\Omega = (0,1]$ be the set. And I have to build a Borel-sigma algebra for $\Omega$. So at first, I have to define the collection of all open intervals in the set $(0, 1]$. Therefore, I have to ask what the meaning of open intervals in $(0, 1]$? What is the collection of all open intervals in the set $(0, 1]$ look like?
My intuition says that the collection of all the open intervals in $(0, 1]$ denoted by $\mathcal{C}$ looks like
$$\mathcal{C}:=\{(a, b) : 0 \leq a < b \leq 1\}\cup \{(a, 1] : 0 \leq a<1\}$$
Please correct me if my notion of open intervals in $(0, 1]$ is wrong.
In general, if you have a topological space $X$ and a subset $Y$ of $X$, you turn $Y$ into a topological space by saying that the open subsets of $Y$ are the sets of the form $U \cap Y$ with $U$ an open subset of $X$.
In your case, the topology on $(0,1]$ is inherited from that on ${\mathbb R}$ and the open subsets of $(0,1]$ are the sets of the form $U \cap (0,1]$ with $U$ an open subset of ${\mathbb R}$.
In particular, this means that $(a,b)$ with $a,b \in [0,1]$ and $a < b$ is an open subset of $(0,1]$ since it is equal to $(a,b) \cap (0,1]$; it also means that $(a,1]$ with $a \in (0,1)$ is an open subset of $(0,1]$ since it is equal to $(a,2) \cap (0,1]$.