Small scattered topological invariants

We present a unified approach to define {dimension functions like} $mathop{{rm trind}}$, $mathop{{rm trind}}_p$, $mathop{{rm trt}}$ and $mathop{{rm p}}$. We show how some similar facts on these functions can be proved similarly. Moreover, several new classes of infinite-dimensional spaces close to the classes of countable-dimensional and $sigma$-hereditarily disconnected ones are introduced. We prove a compactification theorem for these classes.