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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?

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    $\begingroup$ I don’t think your intuition is about open sets; your intuition of “region around a certain point” seems closer to the notion of neighborhood of a point. An open set is a set that is a neighborhood of all its points. $\endgroup$ – Arturo Magidin Apr 6 at 19:34
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    $\begingroup$ A neighborhood of $p$ is any set that includes “a region near $p$”, not merely a set where everything is “a region near $p$”. $\endgroup$ – Arturo Magidin Apr 6 at 19:40
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    $\begingroup$ No, because we require it to happen at all neighborhoods. If you ask who lives “near” me, then depending on your definition of “near” I may mention people who live a block away, or may two hours drive away. Look, you could think of an undefined concept of “close neighborhood” of a point, which is a bit nebulous but intuitively is not just things near by, but only things near by. Then a neighborhood is any set that includes a “close neighborhood”; and then an open set is any set that includes a neighborhood of all of its points. $\endgroup$ – Arturo Magidin Apr 6 at 19:50
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    $\begingroup$ You can define continuity in terms of a basis for a system of neighborhoods. It’s an equivalent formulation. You can define topology in terms of a system of neighborhods for each point. You can also define continuity in terms of special families of open sets (a basis, or even a subbasis). But ultimately, the reason you are having trouble is that you are focusing on precisely the wrong thing. “locality”, “far away”, etc. are metric concepts. Topology does not concern itself with that type of notion, by design. If you insist on trying to fit it into it, you get silliness. Well, yes. $\endgroup$ – Arturo Magidin Apr 6 at 20:28
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    $\begingroup$ By forgetting your metric notion of “locality.” The locality that topology concerns itself with is precisely the locality of neighborhoods of a point, but that is the locality that you insist on rejecting because if it lives inside a metric space then it doesn’t reconcile with the metric notion of locality that you keep trying to bring to the party. $\endgroup$ – Arturo Magidin Apr 6 at 20:35

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