Free exercises in topology


To H. Poincaré


       UPON SOME TOPOLOGICAL IMAGE-PROCESSING MAPPINGS. The geometry of one “thing” into another “thing” works out an essentially wider concept of thing. It’s a kind of rephrasing the whole nearness of it as a set, numbering, computing, processing 1 and 0.

       Let \left\{  X_{i}\right\}  _{i\epsilon N} be some sequence of sharp shafts flickering, interrupting the perfection of the dark in the neighbourhood of x, on half a map intentionally left blank, unfinished.
       The closed surface of objects creases the smoothness of space, fuzzes the crisp path of light.



       THEOREM 1. The collection of all objects forms a covering of empty space, whose intersection is nonempty.

       Proof. The whole space is dense with things, with uncountable, unmeasurable points. H(-oles) are mapped into 0. Objects shrink under the whiteness of multi-layered strings of 1, fill in surrounding ε-open balls. 1 sprouts. Objects aim at replacing the hollowness of zero to enter earthly light. Binary masks perform on dark ROI (Region Of Interest). The extinction axiom is thus arbitrary. Objects intersect other objects along strings of 1.

       There has always to be some flash first for things to inherit the world. This is called restriction of the world to the light and denoted by W_{L}. Under this condition light is homeomorphic to the dark.



       LEMMA 1 (THE CAUCHY-GOETHE SINGULARITY). Light curves on the roundness of dust, on the rough spots of a thin coat of paint. Its many reflections never cease for a lifetime.

       Proof. Intentionally left to the reader.

       Shapes must be observed before they either disappear or fall, collapsing under the weight of the dark. If some space is disconnected, then it is natural to attempt to decompose it into smaller connected pieces (viz. things).


Map bark into skin, gears into nerves, without reference to any Euclidean metric ρ.

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