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Sculptured
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I'm having some trouble understanding the distinction between closed sets, open sets, and those which are neither when the set itself involves there not being a finite boundary. For example, the set { |z - 4| >= |z| : z is complex}. This turns out to be the inequality 2>= Re(z). On the right side it does have a boundary at Re(z) = 2. To the left, up and down it just continues for every possible value of Im(z). Would this even be considered a boundary? Is the set open, closed or neither?
Edit: Nevermind, instead I just want to make sure I'm going down the right road. A set which would extend infinitely would not have a boundary point, as there is no point exterior to the set in that direction, and therefore there isn't a point which is both an exterior and an interior point in the set to be a boundary point. This makes talk about boundaries in this case meaningless and the example above would come out to be a closed set because its only boundary is contained within the set. Correct?
Edit: Nevermind, instead I just want to make sure I'm going down the right road. A set which would extend infinitely would not have a boundary point, as there is no point exterior to the set in that direction, and therefore there isn't a point which is both an exterior and an interior point in the set to be a boundary point. This makes talk about boundaries in this case meaningless and the example above would come out to be a closed set because its only boundary is contained within the set. Correct?
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