(Although I am taking an algebraic topology class, this is not homework;
we have not gotten to this yet.)
Let $\langle X,\mathcal{T}_X\rangle$ be a path-connected Hausdorff space. $\:$ Let $x_0$ be a member of $X$. $\:$ Let $L$ be the set
loops in $X$ with endpoints $x_0$. $\;\;\;$ Define $\;\; \operatorname{concat} \; : \; L\times L \; \to \; L \;\;$ to be loop concatenation.
Define the relation $\:\sim\:$ on $L$ to be homotopicness relative to endpoints. $\;\;$ Define $\mathcal{T}_L$ to
be the compact-open topology on $L$. $\:$ It follows that $\:\sim\:$ is an equivalence relation on $L$.
Define $\;\; [\cdot] \; : \; L \; \to \; L/\sim \;\;$ to be the quotient map.
It follows that for all members $\hspace{.4 in} f_0,f_1,g_0,g_1$ of $L$, $\;\;$ if $\: f_0 \sim f_1 \:$ and $\: g_0 \sim g_1 \:$ then $\operatorname{concat}(f_0,g_0) \sim \operatorname{concat}(f_1,g_1) \:$.
Define $\;\; \star \; : \; (L/\sim) \times (L/\sim) \; \to \; L/\sim \;\;$ by $\;\; [f]\star [g] \: = \: [\operatorname{concat}(f,g)] \;\;$.
It follows that $\langle (L/\sim),\star \rangle$ is a group and that $[t\mapsto x_0]$ is its identity element.
Define $\mathcal{T}_G$ to be the quotient topology on $\: L/\sim \:$ from $\mathcal{T}_L$.
When is $\langle (L/\sim),\star,\mathcal{T}_G \rangle$ a not-necessarily-Hausdorff topological group?
When is $\{[t\mapsto x_0]\}$ closed with respect to $\mathcal{T}_G\:$?
I have convinced myself that both of those happen at least when:
for every point $x$ in $X$, there is an open subsets $U$ of $X$ such that $\: x\in U \:$ and $U$ simultaneously
witnesses local path connectedness and semi-local simply connectedness of $\langle X,\mathcal{T}_X\rangle$.
(because I have convinced myself that $\langle (L/\sim),\mathcal{T}_G\rangle$ is discrete in that case)