Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
Here $|K|$ denotes the underlying topological space. One here should not get confused with the meaning of open face: If $$ F:=[x_0,x_1,\ldots x_k]= \{\sum_{j=0}^{k}\lambda_i x_i:\sum \lambda_i=1,\lambda_i\geq 0\} $$ is a $k$-dimentsional closed face of $K$ then the open face associated to $F$ (it is not open in the subspace topology of $|K|$ in general) is defined as $$ F^{\circ}:=\{\sum_{j=0}^{k}\lambda_i x_i:\sum \lambda_i=1,\lambda_i> 0\}. $$ By definition we will say that a face of dimension $0$ of $K$ ( which is just a point) is an open face of $K$. I hope that the next question is true but I'm note sure how I would prove it
Q1: Let $K'$ be an incomplete finite simplicial complex. Is it always possible to find a finite simplicial complex $A$ such that $|K'|$ deformation retracts to $|A|$ ?
I'm less sure about the positivity of the next question:
Q2: Let $K'$ be an incomplete finite simplicial complex. Does $|K'|$ admit a triangulation (the triangulation won't be finite in general since $|K'|$ is not necessarily compact) ?