To construct the fan start with the
Cantor set which we will call C along the x axis and a point at \left(\tfrac1{2},\tfrac1{2}\right) which we will call p. Join every point in C to p with a straight line. We now have a set that is
connected and becomes disconnected if we remove p. To make the set
totally disconnected when we remove p we need to remove more points. If we look at how the Cantor set C was constructed we see that some points in C like \tfrac1{3} or 1 were the endpoints of
intervals we removed when constructing it and others like \tfrac1{4} are not. We use this to decide which of the points we remove from every line. If the point at the bottom of a line (which will be part of C) is one of the points that was an end point of an interval we remove all coordinates with
irrational y coordinates. Otherwise we remove all points on the line with
rational y coordinates. This set is the Knaster–Kuratowski fan. ==Properties==