The criterion for "EVERY continuous map from $D$ to $[0, 1]$ has a continuous extension to $X$" is that any two disjoint zerosets in $D$ have disjoint closures in $T$. X$. You can find this in Chapter 6 of Gillman and Jerison's classic "Rings of Continuous Functions". They also consider the "local problem" of continuously extending a single map at length in some of the exercises, e.g. given$f:D\rightarrow Y$(not necessarily$Y=[0, 1]$) Exercise 6G characterizes the largest subspace of$X$to which$f$can be continuously extended in terms of$z$-filters. 1 The criterion for "EVERY continuous map from$D$to$[0, 1]$has a continuous extension to$X$" is that any two disjoint zerosets in$D$have disjoint closures in$T$. You can find this in Chapter 6 of Gillman and Jerison's classic "Rings of Continuous Functions". They also consider the "local problem" of continuously extending a single map at length in some of the exercises, e.g. given$f:D\rightarrow Y$(not necessarily$Y=[0, 1]$) Exercise 6G characterizes the largest subspace of$X$to which$f$can be continuously extended in terms of$z\$-filters.