Let $\mathcal{U} = \{ U\_i \; |\; i\in I \}$ be an open covering of $X$‎. Spanier defined $\pi (\mathcal{U}‎, ‎x)$ to be the subgroup of $\pi\_1 (X‎, ‎x)$ which contains all homotopy classes having representatives of the following type‎: $
‎\prod\_{j=1}^{n}u\_j \*v\_j \* u^{-1}\_{j}‎,
‎$
‎where $u\_j$'s are paths (starting at the base point $x$) and each $v\_j$ is a loop inside one of the neighbourhoods $U\_i \in \mathcal{U}$‎. 


‎If an open cover $‎\mathcal{U}$ is a refinement of an open cover $‎\mathcal{V}$‎, then $\pi (‎\mathcal{U}‎, ‎x) \subset \pi (‎\mathcal{V}‎, ‎x)$‎. 


My question is that:


If $[f][g]\in \pi (‎\mathcal{U}‎, ‎x)$ for $[f],[g]\in \pi\_1 (X,x)$, then is there any refinement $\mathcal{V}$ of $\mathcal{U}$ so that $[f]\in \pi (\mathcal{V},x)$?



Concerning the Spanier group relative to an open cover
No. Let $X\_1=X\_2=X\_3=S^1$ be copies of the unit circle and consider $X\_1\vee X\_2\vee X\_3$ with wedge basepoint $x$. Let $\gamma\_i$ be a loop traversing $X\_i$ whose homotopy class generates $\pi\_1(X\_i,x)$. Construct $X$ by attaching a 2-cell to $X\_1\vee X\_2\vee X\_3$ by the attaching loop $\gamma\_1\ast\gamma\_2\ast \gamma\_{3}^{-1}$. 


Consider an open cover $\mathscr{U}$ of $X$ consisting of


* the interior of the attached 2-cell,
* an open neighborhood $U$ of $X\_3$ that deformation retracts onto $X\_3$ (i.e. $X\_3$ with a partial collar),
* Small simply connected sets in $X$ whose union contains $X\_1\vee X\_2$


Notice that $U$ is the only non-simply connected set in $\mathscr{U}$ and so $\pi\_1(\mathscr{U},x)$ is the normal subgroup of $\pi\_1(X,x)$ generated by $[\gamma\_3]$. In particular, $[\gamma\_1][\gamma\_2]=[\gamma\_3]\in \pi\_1(\mathscr{U},x)$ but $[\gamma\_1]\notin \pi\_1(\mathscr{U},x)$. It follows that if $\mathscr{V}$ is a refinement of $\mathscr{U}$, then $[\gamma\_1]$ can't lie in $\pi\_1(\mathscr{V},x)$ either since it's a subgroup of $\pi\_1(\mathscr{U},x)$.


