Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds
Abstract
Let (X,τ) be metrizable and let F={a1,…,ak}⊆ X, where 2≤ k<∞. We represent all metrizable topologies σ⊆τ agreeing with τ on X∖ F by compatible systems of continuous gauges si:X→[0,1] with si−1(0)={ai}. The condition infXmax{si,sj}>0 for i≠ 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.
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BibTeX
@misc{kalvan2026finitepoint,
title = {{Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds}},
author = {Ahmad ja'afari kalvan and Ehsan Shahoseini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20514},
eprint = {2609.20514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20514},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds
AU - Ahmad ja'afari kalvan
AU - Ehsan Shahoseini
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20514
UR - https://arxiv.org/abs/2609.20514
AB - Let (X,τ) be metrizable and let F={a_1,…,a_k}⊆ X, where 2≤ k<∞. We represent all metrizable topologies σ⊆τ agreeing with τ on X∖ F by compatible systems of continuous gauges s_i:X→[0,1] with s_i^−1(0)={a_i}. The condition inf_Xmax{s_i,s_j}>0 for i≠ 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.
ER -