Suppose $X$ is a compact space. In particular $X$ is totally bounded and there exists $x_1,..,x_n$ such that $$ X = \bigcup_{i=1}^n U(x_i, \epsilon) $$ where $U$ is the Open Ball centered at $x_i$ with radius $\epsilon$ w.r.t the metric on $X$ In the case where $X$ is a set of certain continuo...