Let $ X $ be a compact metric space and let $ H(X) $ be the collection of all homeomorphisms on the space $ X $ with the $ C^0 $-metric \begin{equation*} d_0(f,g)=\max_{x\in X}d(f(x),g(x))+\max_{x\in X}d(f^{-1}(x),g^{-1}(x)). \end{equation*} Let $ H_m (X) $ be the collection of all semigroups on the space $ X$ with finite set of generators $ \{id, g_1 , \ldots,g_m\} $, where $ g_i \in H(X), i=1,\ldots,m$. Given two semigroups $ F,G\in H_m (X) $ generated by the finite family $ F_1=\{id,f_1, \ldots, f_m\}$ and $G_1=\{id,g_1, \ldots, g_m\}$, respectively.
Defined the $ C^0 $-metric on $ H_m (X) $ as \begin{equation*} D_0(F,G)=\max_{1\leq i\leq m}d_0(f_i , g_i ). \end{equation*} Semigroup $G$ generate by the finite family $G_1=\{id,g_1, \ldots, g_m\}$, is called commutative semigroup if $g_i\circ g_j=g_j\circ g_i$ for all $1\leq I, j\leq m$.
In my research, I must to work with commutative semigroups.
Q. Let $H'_m(X)$ be the set of all commutative semigroups on $X$. What can about topology on $H'_m(X)$? Can I study it as a metric space or as a subspace of $H_m(X)$?
Please help me to know it.