I am working on Exercise 4.1.11 in Hatcher's Algebraic Topology text, which is as follows: Show that a CW complex $X$ is contractible if it is the union of an increasing sequence of subcomplexes $X_1 \subset X_2 \subset \cdots$ such that each inclusion $X_i \hookrightarrow X_{i+1}$ is nullhomoto...

This is how the Wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and multiplication, and contains the multiplicative identity of $R$. When you follow the link for the subring...

As your counterexample shows, the first statement is incorrect. But there is a one character fix: require that the subset contains $\,{-}1\,$ vs. $\,1.\,$ Alternatively, require it to also be closed under negation (additive inverses). Perhaps they meant $\rm\,S\,$ is closed under subtraction (vs...