One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation
Abstract
Let (X,τ) be metrizable and let a∈ X. We give a constructive account of metrizable topologies σ⊆τ that agree with τ on X∖{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≤ d that agrees with d on a common neighborhood of each point other than a. A prescribed countably infinite closed discrete set {xn:n∈N}⊆ X∖{a} can be made to satisfy p(a,xn)≤λn for any positive null sequence (λn). 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.
Related
- Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds
- Metric preservation under approximate algebraic laws
- Topological preservation under aggregation of quantale-valued metric spaces
- Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes
- Presenting Point-Glued Self-Similar Objects as Final Coalgebras in Metric Spaces via Completion: A Categorical Hutchinson Theorem
BibTeX
@misc{kalvan2026onepoint,
title = {{One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation}},
author = {Ahmad ja'afari kalvan and Ehsan Shahoseini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13827},
eprint = {2609.13827},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13827},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation
AU - Ahmad ja'afari kalvan
AU - Ehsan Shahoseini
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13827
UR - https://arxiv.org/abs/2609.13827
AB - Let (X,τ) be metrizable and let a∈ X. We give a constructive account of metrizable topologies σ⊆τ that agree with τ on X∖{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≤ 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∈N}⊆ X∖{a} can be made to satisfy p(a,x_n)≤λ_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.
ER -