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Perfectly Normal Spaces
A perfectly normal topological space is one where disjoint closed sets can be precisely separated by a continuous function. That is, there is a continuous function from the topological space to the closed unit interval such that the fiber of 0 is the first closed set, and the fiber of 1 is the second closed set.
All metrizable spaces are perfectly normal, the dist function provides a means of explicitly writing down the separating function. So, in particular, the Euclidean plane with the standard Euclidean metric is perfectly normal.
Given two disjoint closed polygonal figures we can apply the separating function to the plane and get a visual as to what it may look like. We do this by coloring a point (x, y) in the plane a color between red and blue based on the value of the function. The first closed polygonal figure is precisely red, and the second closed polygonal figure is precisely blue. The colors vary continuously across the plane, giving a rainbow gradient.
Tools for generating rasterized images of this function are provided for any collection of disjoint polygonal figures.
Below is an example depicting two closed sets that are each the finite union of closed polygonal regions. The characters spell out potato, or tu dou, in simplified Chinese (a running joke from my topology course).
