Approximation relations on the posets of pseudometrics and of pseudoultrametrics

Abstract

We show that non-trivial "way below" and "way above" relations on the posets of all pseudometrics and of all pseudoultrametrics on a fixed set $X$ are possible if and only if the set $X$ is finite.

Linked bicontinuity of posets of compact ultrapseudometrics

“…Sections 4 and 5 consider two dual notions, of approximation from below and from above, in posets of compact ultrapseudometrics (in subsets of these posets mostly, to be more precise). We show why, for the approximation relations to be reasonably good, we should restrict our attention to ultrapseudometric only (instead of all pseudometrics), and, moreover, to compact ones (even local compactness is not sufficient) [8]. While results of Section 4 have already been published in [9], Section 5 presents new findings with complete proofs.…”

“…If X is finite, the description is simple. Proposition 2 (Proposition 1, [8]). For pseudometrics d 0 and d 1 on a finite set X the following statements are equivalent:…”

“…In [8] rather discouraging results were obtained on the approximation relations in the poset (P s(X), ⩽). If X is finite, the description is simple.…”

“…This paper serves two purposes: it summarizes the previous research [8][9][10] and presents new results that finally allow to get a more or less complete view what the posets in question are.…”

Compact ultrapseudometrics and inverse systems

“…Як бачимо, частковому порядку на отриманiй множинi класiв строго iзоморфних R C -спектрiв бракує корисних властивостей. Є два шляхи до "полiпшення" цiєї множини -перехiд до класiв строго iзоморфних конусiв з фiксованою вершиною, яка є нульвимiрним компактом злiченної ваги [4], або послаблення вимог до морфiзмiв -вiдмова вiд строгостi, тобто розгляд складнiшого поняття морфiзму, прийнятого у теорiї зворотних спектрiв [9], i вивчення фактор-об'єктiв [3]. Обидва шляхи ведуть до цiкавих результатiв, яким будуть присвяченi наступнi публiкацiї.…”