Let $C$ be the Cantor set, and $\omega$ the discrete space of integers $\{0,1,2,...\}$.

My conjecture:

(1) For each $n<\omega$ let $f_n:C\to C$ be a continuous function (possibly not onto). Let $$X=(\omega\times C)/\sim.$$ This is the quotient of $\omega\times C$ under the relation $\sim$: $$\langle n,c\rangle\sim\langle m,d\rangle\iff(\exists l\geq n,m)(f_l\circ ...\circ f_n(c)=f_l\circ ...\circ f_m(d)).$$ Then $\dim(X)=0$.

Is it true? I have not found it anywhere, despite doing a Google and looking in Dimension Theory texts.

The space $X$ is sometimes called a *direct limit* https://en.wikipedia.org/wiki/Direct_limit.

By $\dim(X)=0$, I mean $X$ has a basis of clopen sets (*small inductive dimension* is $0$).

More generally, I suppose one could ask if the direct limit of zero-dimensional spaces has dimension zero. But letting each factor be *equal* to the Cantor set potentially makes things easier.

What leads me to believe the statement is true, is two extreme cases of $f_n$'s. If each is constant, then $X$ is a singleton. And if each is the identity, $X$ is an increasing union of $\omega$-many Cantor sets. In this regard, the statement seems to be a generalization of the well-known "countable sum theorem" for (separable metric) spaces:

(2) If a separable metric space $X$ can be represented as the union of a sequence $F_0 , F_1, ...$ of closed zero-dimensional subspaces, then X is zero-dimensional.

I'm unsure if this is question is "research-level", so feel free to move to MSE if appropriate.

EDIT: Just to be clear, I am asking for either a reference or a proof of (1). In the meantime I'll try to prove it by mimicing the proof of the (2), which can be found on page 20 (item 1.3.1) of this book: http://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf