I have such a problem.
Let $K,L$ be simplicial complexes, where $K$ is connected, countable and locally finite. In particular we can express $$K=\bigcup_{i=0}^\infty K_i,$$ where $K_0$ is some full and finite subcomplex (in particular compact) and $K_{i+1}$ contains all the vertices of $K$ that are joined by an edge with a vertex of $K_i$. From locally finitness every $K_i$ is a finite, full subcomplex. Moreover, by construction we have $|K_i|\subset |K_{i+1}|$.
Now let $f,g:|K|\to |L|$ be two continuous maps that $f|_{|K_i|}\simeq g|_{|K_i|}$ for every $i=0,1,...$.
Does it imply that $f\simeq g$? We may assume that $f$ and $g$ are simplicial.
