I'm studying the entropy for flows via reparametrizations. Let (X,d) a compact metric space. For a closed interval $I$ which contains the origin, a continuous map $\alpha: I \rightarrow \mathbb{R}$ is called a reparametrization if it is a homeomorphism onto its image and $\alpha(0)=0$. The set of all such reparametrizations on $I$ is denoted by $\operatorname{Rep}(I)$. For a flow $\phi$ on $X, x \in X, t \in \mathbb{R}^{+}$and $\varepsilon>0$, we set
and call $B(x, t, \varepsilon, \phi)$ a $(t, \varepsilon, \phi)-$ ball or a reparametrization ball in $X$.
I want to prove that all the reparametrization balls are open sets. I would appreciate some comments or suggestions.