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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) ?

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  • $\begingroup$ Can I ask for an example of an incomplete finite simplicial complex? I work better with examples. $\endgroup$
    – user62675
    Jan 26, 2015 at 22:16
  • $\begingroup$ Sure, for example take the barycentric subdivision of a 2 simplex (a triangle plus its inside). Then remove the barycenter which is an open 0-simplex. Then this incomplete finite simplicial complex deform retracts to a triangle (a 3-cycle of 1-simplices). $\endgroup$ Jan 26, 2015 at 22:56
  • $\begingroup$ My first guess is that the answer to both questions is yes. About Q1, why can't you obtain A by taking a barycentric subdivision of K and removing all simplices that intersect Y? About Q2, can't you prove that |K'| is homeomorphic to a polyhedron as in chapter 1 of Rourke-Sanderson and hence admits a triangulation? $\endgroup$
    – skupers
    Jan 26, 2015 at 23:18
  • $\begingroup$ Hi Skupers, concerning your approach to Q1, it seems intuitively correct but nevertheless feel shaky with this kind of argument. It seems to me that whenever you remove an open face, you create a "hole" inside $|K'|$ which you can enlarge and push on the sides of the remaining complete simplices of |K'|. But again I feel a bit uncomfortable about the rigour of this kind of argument $\endgroup$ Jan 27, 2015 at 12:42

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