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July 1, 2016 / porton

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.

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2 Comments

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  1. porton / Jul 1 2016 20:36

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

  2. porton / Jul 17 2016 20:48

    Oh well, wrong. See my manuscript instead.

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