acyclic

A continuum X is said to be acyclic provided that each mapping from X into the unit circle $\mathbb{S}^1$ is homotopic to a constant mapping, i.e., for all mappings $f: X \to \mathbb{S}^1$ and $c: X \to \{p\} \subset \mathbb{S}^1$ there exists a mapping $h: X \times [0,1] \to \mathbb{S}^1$ such that for each point $x \in X$ we have h(x,0) = f(x) and h(x,1) = c(x).