What is an open set in a topological space?

There is no such thing as an "open set," only a open subset of a topological space. "The singleton $\{ 1 \}$ is an open set" is not a meaningful statement; the meaningful statement is whether the singleton $\{ 1 \}$ is an open subset of some topological space containing $1$. For example it is not an open subset of $\mathbb{R}$, or of the closed interval $[0, 1]$ (with their usual topologies). However, it is an open subset of itself, considered as a one-point space (which has a unique topology).

Sets in a topology are open by definition; that's what "open" means.

I suggest that when starting to learn topology you completely ignore the ordinary English meaning of the word "open" and just work with the axioms abstractly for awhile. It's not the most fun or intuitive way to do it but at least you aren't letting preconceptions get in the way. Also, I don't really recommend this as a way to better understand calculus; you'd be better off picking up a textbook on real analysis.

The whole point of having a general topology is that you get to define which sets are and aren't "open", to make the rules of the game, and then get to see what that does and how things are different in the "world" so created versus the usual real numbers. If we take the idea that an open set "doesn't contain its own boundary", which is what you are after but how I originally heard it phrased, being able to define open sets to be whatever you want them to be (so long as you meet the rules for how they must be structured under union and intersection) means, in effect, you get to define what constitutes a "boundary" and what doesn't. You get to make what is and isn't an "end point".

To see why that has an impact, note that the only reason that $0$ and $1$ are "boundaries" of $(0, 1)$ is because of the ordering on the reals, which ensures that $0 < x < 1$ whenever $x \in (0, 1)$, and also, there's nothing in between 0 and 1 and the set $(0, 1)$, i.e. no points $y$ such that $0 < y < x$ for every $x \in (0, 1)$, and similarly for $1$.

But suppose we re-ordered the reals, so that both points $0$ and $1$ came before the points we consider to be in $(0, 1)$ (in the usual definition.). E.g. suppose we ordered the reals to look like

$$(\text{stuff}) < 0 < 1 < 2 < (\text{stuff}) < (\text{the numbers in $(0, 1)$}) < (\text{more stuff})$$

Now, suddenly, $(0, 1)$ no longer has boundary points $0$ and $1$. So there is no absolute notion of a "boundary point". It depends on the order, and we just redefined what the boundary was by redefining the order.

And topology is even more flexible than that. And orders are just one source, but far from the only one, of topologies.