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- Mar 11th 2013, 03:24 PMredtdcS
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- Mar 11th 2013, 03:34 PMILikeSerenaRe: Sketching regions in the complex plane
- Mar 11th 2013, 03:40 PMPlatoRe: Sketching regions in the complex plane
- Mar 11th 2013, 04:19 PMHallsofIvyRe: Sketching regions in the complex plane
|z- 3i| is the distance from z to 3i and |x- 2| is the distance from z to 2 so S consists of points that are equally distant from 3i and 2. Geometrically, the set of all points equally distant from point P and Q is the perpendicular bisector of the segment PQ.

Algebraically, taking z= x+ iy, |z- 3i|= |z- 2| is the same as $\displaystyle \sqrt{(x^2+ (y- 3))^2}= \sqrt{((x- 2)^2+ y^2}$ which is the same as $\displaystyle x^2+ (y- 3)^2= (x- 2)^2+ y^2$.

As for whether it is open or close, what topology are you using?