Intuition on open set

Question

I have some basic knowledge on topology. Currently my intuition on open set roughly comes from the concept of open ball in metric space, that is to say, an open set is 'a region around a certain point'. This intuition has been fine for me to continue my study, but I find it not satisfactory for some reasons.

The first reason is because of arbitrary union. The concept of continuity at a point $p$ in any analysis book is roughly like, for some region $V$ around $f(p)$ there exists a region $U$ around $p$, such that $f(U) \subseteq V$. If we replace the word 'region' with 'open set', it seems to be that the concept of open set fits here perfectly. However since open set allows arbitrary union, it's perfectly valid that an open set $U \ni p$ consists of two disconnected open sets that are far away in space. It feels weird that when we are talking about the contiunity at a specific point, we drag in some arbitrary region somewhere far away into our arguments. Say $(-1, 1) \cup (100, 101)$ is an open set in $\mathbb{R}$, when we consider continuity at $0$, we only should be taking $(-1, 1)$ into consideration, and it has nothing to do with $(100, 101)$. But since we are talk about open set in general, we have to take $(100, 101)$ implictly into our arguments. Since open set can be arbitrary union, it doesn't really fit into the idea of 'a region around some point', but continuity seems to me relies on this idea. It's great that open set formulation works (since $f(U_1) \subseteq f(U_1 \cup U_2) \subseteq V$), but I hope there's a better reconciliation between the concept of open set (with arbitrary union) and region around a point.

The second reason is just that if I think about open set as 'region around a point', I seem to be missing all the subtleties of openness, closedness and boundary. Because we can easily replace open set with closed set and say a closed set is also a 'region around a point', thus this idea is apparently too coarse. All these concepts are kind of interrelated, but I want to have a starting point in terms of the intuition. For example boundary point is considered as a point such that any region around it no matter how small will intersect with both $S$ and $S^C$. But this intuition still relies on the concept of open set, so to me it feels like to be begging the original question on the intuition of open set.

There're some threads explaining that an open set is actually a semi-decidable property. It feels like that direction is promising, but I don't see very clearly how that idea materialise under the interpretation of point space.

Are there better intuitions for open sets?