When can I assume that an open interval is an open set in topology?

Question

I am confused as to what axioms are at my disposal. I am using Viro, Yvanov, Netsvelaev Elementary Topology Problem Book and it uses an IBL-Approach. The definition I have so far are "elements of a topological structure $\Omega$ of a given set $X$ are open sets" and "a set is closed iff its complement is open". Then I was asked to show that the half-open interval $(0,1]$ is neither open or closed in $\Bbb{R}$. The hint provided asked the reader to assume that $(0,1]$ is open and consider $(0,1]=\bigcup_i(a_i,b_i)$. Though I have successfully arrived at a solution, my issue with this solution is why am I allowed to immediately assume that $\forall i,$ $(a_i,b_i)$ is open? Obviously, it is open but how do I show this with just the two definitions provided.

Answer 1

It only makes sense to ask for a proof that $(0,1]$ is neighter open or closed in $\mathbb R$ if it is clear with which topology $\mathbb R$ is equipped in this situation.

The fact that this question is placed on your plate indicates that this is indeed the case (so check this yourself).

The hint further makes clear that you are dealing with what we call the "usual" topology on $\mathbb R$ and a set $U\subseteq\mathbb R$ belongs to this topology if and only if it can be written as a union $U=\bigcup_{i\in I}(a_i,b_i)$.