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I spent a couple of hours today trying to prove the following:

Let $L$ be a subcomplex of a simplicial complex $K$. Let $U_L$ be the union of the relative interiors of the relative interiors of all simplices in $K$ that have at least one vertex in $L$.

(a) Show that $U_L$ is an open set in $|K|$ (the geometric realization of $K$) containing $|L|$.

(b) Assume that, each simplex in $K$ intersects $L$ in exactly one face (possibly empty) of the simplex. By finding a canonical retraction, and a canonical homotopy from the identity to the composition of the inclusion and the retraction, show that $|L|$ is a deformation retract of $U_L$.

For (a), I was thinking that where $v$ is a vertex of $|L|$, $$U_L = \cup_{v \in |L|} Star(v),$$ where $Star(v)$ is the star of $v$ in $|K|$; if this is (close to) correct, then since I have proven already that each star is open in $|K|$ and the union of all vertex stars covers $|K|$, (a) should follow easily from there. I am not sure if this approach is right, so any constructive feedback would be awesome!

I have had more trouble building momentum on (b). I am hoping that having an appropriate retraction will yield a homotopy in a simple enough way. I am wondering if anyone visiting the site would be up for either giving suggestions towards proving this result or for outlining a proof of this result. Thanks in advance for constructive feedback.

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Isn't the $K=\text{union of two edges of }\partial\Delta^2$ and $L=\text{boundary of the remaining edge}$ a counterexample for b) ? – Stefan Hamcke Jan 13 '15 at 17:26

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