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Consider this line: y=2x, x is any real number

According to a Finnish math book, this line is "closed on a plane". I tried to look up what it means to be closed or open on a plane and I couldn't really find a specific definition anywhere. Here's something seemingly related. According to that source, a close boundary is confined to a finite surface or volume. I don't know how you take the surface of a line. Is it zero? Does this make it "finite", even though the line extends to infinity in both directions?

Can you provide examples to illustrate what it means for something to be "open/closed on a plane"?

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The concepts "open" and "close" belong to topology. A set it's closed if it's equal to its closure. The closure of a set is the set of points having for any neighbourhood (neighbourhood=any open set containing the point) a not empty intersection with the initial set. You can consider a closed set intuitively as a set containing it's border. The contrary for an open one.

A line in the plane is a closed set because it is its own border (intuitively, but the definition works). An open set in the plane is e.g. $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ being the border $\{(x,y)\in\mathbb R^2|x^2+y^2=1\}$ not belonging to $A$

But in the link you give, the concepts "open" and "closed" are used with a different meaning. The finish book is referring to the topological, the link to analysis.