Earlier I’ve conceived an algebraic formula to characterize whether a quasi-uniform space is normal (where normality is defined in Taras Banakh sense, not in the sense of underlying topology being normal). That my formula was erroneous.

Today, I have proved another formula for this (hopefully now correct):

Theorem An endoreloid $latex f$ is normal iff $latex \mathsf{Compl} (\mathsf{FCD}) f^{- 1} \circ \mathsf{CoCompl} (\mathsf{FCD}) f \sqsubseteq \mathsf{CoCompl} (\mathsf{FCD}) f \circ (\mathsf{FCD}) f.$

The above formula also applies to any quasi-uniformity $latex f$.

The proof of the theorem is currently available in this PDF file.

Read my free ebook to understand formulas like this.

2 thoughts on “Reexamined: Normal quasi-uniformity elegantly defined

  1. From this formula it also follows that normality is determined by the underlying proximity and does not need particular uniformity.

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