On the metric reflection of a pseudometric space in ZF

A set-theoretic definition of cell types with an algebraic structure on gene regulatory networks and application in annotation of RNA-seq data

“… is a pseudo-metric space. Additional information on the topological space, open ball, and pseudo-metric space is provided in the Supplemental experimental procedures ( Herrlich and Keremedis, 2015 ; Manetti, 2015 ).…”

“…Supposing that, when the definition of cell classes coincides with the GRN (that is, ; Figure 3 E), . Hence, when we define , where is the fiber of , is defined as a metric space because the metric reflection (the equivalence relation ) has been introduced in the pseudo-metric space ( Herrlich and Keremedis, 2015 ). Therefore, in this case, the Hamming distance is suitable for cell classes as well.…”

On preimages of ultrafilters in ZF

“…The principle PUU (: For every infinite set X, Y , for every onto function f : X → Y , for every ultrafilter F of Y , f −1 (F ) = {f −1 (F ) : F ∈ F } extends to an ultrafilter of X) was introduced in [2] in order to prove:…”

“…(A) For every pseudometric space X, if X is ultrafilter compact then so is its metric reflection X * . The question whether PUU is necessary for the proof of (A) was left unanswered in [2]. The latter question leads to the following additional two:…”