Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are *contiguous* if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous simplicial maps have homotopic geometric realisations: $|f|\simeq |g|$.

Let $K$, $L$ be simplicial complexes and let $(f_\phi\colon K\to L)_{\phi\leq \alpha}$ be a transfinite sequence of simplicial maps such that:

- $f_\phi$ is contiguous to $f_{\phi+1}$ for all $\phi<\alpha$;
- for each limit ordinal $\psi$ and each vertex $v\in K$ there is some $\beta<\psi$ such that $f_\phi(v)=f_\beta(v)=f_\psi(v)$ for each $\beta\leq \phi<\psi$.

Is there a homotopy between $|f_0|$ and $|f_\alpha|$?

If $\alpha$ is countable, we can embed it into $[0,1]$ and construct a homotopy by gluing homotopies between consecutive maps in the sequence. (I believe this should be continuous.) But what happens for arbitrary $\alpha$?

Is there some general theorem that answers this kind of questions? For a sequence of spaces instead of a sequence of maps I would use the Whitehead theorem, but see Whitehead for maps.