Open subset topology
In all but the last section of this wiki, the setting will be a general metric space (X,d).(X,d).(X,d). Those readers who are not completely comfortable with abstract metric spaces may think of XXX as being Rn,{\mathbb R}^n,Rn, where n=2n=2n=2 or 333 for concreteness, and the distance function d(x,y)d(x,y)d(x,y) as being the standard Euclidean distance between two points.
So the intuition is that an open set is a set for which any point in the set has a small "halo" around it that is completely contained in the set. The idea is that this halo fails to exist precisely when the point lies on the boundary of the set, so the condition that U UU is open is the same as saying that it doesn't contain any of its boundary points. With the correct definition of boundary, this intuition becomes a theorem.
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