One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation

Let $(X,\tau )$ be metrizable and let $a\in X$. We give a constructive account of metrizable topologies $\sigma \subseteq \tau$ that agree with $\tau$ on $X\setminus \{a\}$. Applying Hausdorff’s classical metric collapse construction, for every noncompact $(X,\tau )$ 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 $\{x_n:n\in \mathbb{N}\}\subseteq X\setminus \{a\}$ can be made to satisfy $p(a,x_n)\le {\lambda }_n$ for any positive null sequence $({\lambda }_n)$. 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.