Compactly supported locally constant function ($C_c^{\infty }(X)$)

The space $C_c^{\infty }(X)$ consists of locally constant complex functions with compact support. On a locally profinite group, finite compact-open coset covers of the support show that every such function is fixed by one common compact-open subgroup under right translation. The symbol infinity describes local constancy in this context, not real differentiability.