arXiv subjects

Ahmad ja'afari kalvan

Publications and source records attributed to Ahmad ja'afari kalvan.

2 recordsLinked to original sources

arXiv2026

Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds

Let $(X,τ)$ be metrizable and let $F=\{a_1,\ldots,a_k\}\subseteq X$, where $2\le k<\infty$. We represent all metrizable topologies $σ\subseteqτ$ agreeing with $τ$ on $X\setminus F$ by compatible systems of continuous gauges $s_i:X\to[0,1]$ with $s_i^{-1}(0)=\{a_i\}$. The condition $\inf_X\max\{s_i,s_j\}>0$ for $i\ne j$ is equivalent to both Hausdorffness and metrizability of the prescribed gauge topology. A normalized product map into the standard simplex gives an explicit metric; its triangle inequality follows from a simplex slack inequality. This metric is complete whenever the auxiliary bounded compatible metric is complete. For two compatible systems, their coordinatewise minimum describes the intersection topology. It is compatible exactly when the two coarsenings have a common Hausdorff lower bound; in that case the intersection is metrizable and is their meet. Otherwise every common lower topology is non-Hausdorff. A closed-discrete construction produces such an obstructed pair for every noncompact metrizable space and every finite exceptional set with at least two points. Consequently, for these exceptional sets, the family is downward directed, or is a lattice, if and only if $(X,τ)$ is compact, in which case it consists only of $τ$.

math.GN↗

arXiv2026

One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation

Let $(X,τ)$ be metrizable and let $a\in X$. We give a constructive account of metrizable topologies $σ\subseteqτ$ that agree with $τ$ on $X\setminus\{a\}$. Applying Hausdorff's classical metric collapse construction, for every noncompact $(X,τ)$ and every compatible metric $d$ we obtain a strict coarsening with a metric $p\le d$ that agrees with $d$ on a common neighborhood of each point other than $a$. A prescribed countably infinite closed discrete set $\{x_n:n\in\N\}\subseteq X\setminus\{a\}$ can be made to satisfy $p(a,x_n)\leλ_n$ for any positive null sequence $(λ_n)$. The resulting metric is greatest among the metrics dominated by $d$ that satisfy these bounds, and is complete whenever $d$ is complete. We exhibit its realization as a classical metric quotient. We also represent all localized metrizable coarsenings by continuous scalar gauges using a standard cone metric. Inclusion is expressed by the cofinal comparison of sublevel sets familiar from extension-trace theory, while pointwise maximum and minimum realize finite joins and meets. A closed-discrete criterion detects strictness. Standard preservation results for Borel structure, complete metrizability, and Polishness, together with function-space and local-field examples, complete the account.

math.GN↗