Remember, topologies are just glorified semi-lattices. If you have two semi-lattices X and Y, and a monotone function f from X to Y then an element a of X is a sufficient factor for b in Y if for any refinement of X W, refinement of Y Z and monotone function f': W -> Z that extends f, for any element w of W, w subs a => f'(w) subs b. Likewise an element a of X is a necessary factor for b in Y if for any refinement of X W, refinement of Y Z and monotone function f': W -> Z that extends f, for any element w of W, w subs a <= f'(w) subs b. An element a of X is a determining factor for b in Y if it is a necessary and sufficient factor. The map f is factorable if every element of Y has a determining factor in X. This means that there exists a function f*: Y -> X. What it means in topology for a map F: X to Y to be continuous is that the induced map f = cl o image_F, from the closed sets of X to the closed sets of Y is a factorable map.

Play today!

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