I am confused about the following question:
Let $K_{\bullet}$ be a simplicial complex. Let $( \sigma, \tau )$ be a pair of its simplices with $\sigma$ the only codimension $1$ coface of $\tau$.
Prove that $L_{\bullet} = K_{\bullet} \setminus \{ \sigma, \tau \}$ is a simplicial complex of the same homotopy type.
I am confused because I am unsure how $\sigma$ can be the only codimension $1$ coface of $\tau$.
Is it not the case that if $S = \{ v_0, ..., v_n \}$ is a simplex, it must have $n+1$ codimension $1$ faces; $ S \setminus \{v_i\}$ where $ i \in \{0, ..., n\}$
Any further hints on how to proceed would be very helpful; I am aware of basic Algebraic topology and was thinking this may be an application of the simplicial approximation theorem?