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)$ ?