Let $X$ be an extremally disconnected compact Hausdorff space *with no open points*, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D\_f$ be the linear span of the functions of the form $$u\to \gamma(u) f(\sigma(u))$$ where $\sigma:X\to X$ is a homeomorphism, and $\gamma:X\to\mathbb{C}$ is continuous with $|\gamma|=1$.



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> **Question:** What is an example of $X$ and $f$ so that $D\_f$ isn't dense in $C(X)$?
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Subspaces generated by the orbits of the group of isometries on $C(K)$
Take two extremally disconnected spaces $Y$ and $Z$ without open points and with different cardinality (say $Z$ is the one with larger cardinality). Define $X$ as the disjoint union of $Y$ and $Z$ and let $f$ be the indicator function of $Y$.


For every sequence $(g\_n)$ in $D\_f$ the union of the supports $\{g\_n \not= 0\}$ is, for cardinality reasons, a proper subset of $X$. So $(g\_n)$ cannot approximate the constant $1$-function on $X$.


