📚 Recommended Mathematics Books
Topology (Munkres) | General Topology (Engelking) | Counterexamples in Topology | Rudin's AnalysisAs an Amazon associate, I earn from qualifying purchases.
This is my first attempt to define micronization.
Definition Let $latex f$ is a binary relation between sets $latex A$ and $latex B$. micronization $latex \mu (f)$ of $latex f$ is the complete funcoid defined by the formula (for every $latex x \in A$)
$latex \left\langle \mu (f) \right\rangle \left\{ x \right\} = \bigcap \left\{
\uparrow^B \left( f x \setminus f y \right) \hspace{1em} | \hspace{1em}
\left( x ; y \right) \in f \right\}. $
\uparrow^B \left( f x \setminus f y \right) \hspace{1em} | \hspace{1em}
\left( x ; y \right) \in f \right\}. $
Conjecture If $latex f$ is a strict partial order, $latex S^{\ast} (\mu (f)) = f$.
The idea of micronization is that it transforms a “global” relation (such as a strict partial order) into a “local” space (something like a topology).
This my definition probably can be generalized for funcoids instead of binary relations.
🔬 Advanced Mathematics References
- Sheaves in Geometry and Logic
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- Stone Spaces
- Algebraic Topology (Hatcher)
- Concrete Mathematics
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