Topological fundamental groups of locally finite infinite configuration spaces and infinite braids

We study the topological fundamental groups of the locally finite infinite ordered configuration space $Con{f}_{\mathrm{\infty }}^{lf}(\text{C})$ in the plane and the homotopy quotient of $Con{f}_{\mathrm{\infty }}^{lf}$ by the canonical action of the infinite permutation group $\text{Aut}(\mathbb{N})$: $$H^{lf}(\mathrm{\infty }):={\pi }_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(Con{f}_{\mathrm{\infty }}^{lf}(\text{C}),\tilde{\mathbb{N}}),B^{lf}(\mathrm{\infty }):={\pi }_1^{\mathrm{t}\mathrm{o}\mathrm{p}}\text{\hspace{0.17em}⁣}(Con{f}_{\mathrm{\infty }}^{lf}(\text{C})\text{\hspace{0.17em}⁣}\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}\text{Aut}(\mathbb{N}),[e_0,\tilde{\mathbb{N}}])\mathrm{.}$$ We prove that $H^{lf}(\mathrm{\infty })$ and $B^{lf}(\mathrm{\infty })$ are non-discrete and complete topological groups. A main structural theorem identifies $H^{lf}(\mathrm{\infty })$ with a canonical locally finite inverse-limit model built from finite pure braid groups, and we construct a complete left-invariant ultrametric compatible with the quotient topology from the loop space of $\text{Conf}$. The direct limit of finite pure braid groups admits a dense embedding into $H^{lf}(\mathrm{\infty })$, and we show that $H^{lf}(\mathrm{\infty })$ is the Raĭkov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into $B^{lf}(\mathrm{\infty })$ and is dense in the finitary subgroup $B_{\mathrm{f}\mathrm{i}\mathrm{n}}^{lf}(\mathrm{\infty })\subseteq B^{lf}(\mathrm{\infty })$.