Reexamined: Normal quasi-uniformity elegantly defined

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 f is normal iff \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 f.

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

Read my free ebook to understand formulas like this.


  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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