Finite simplicial open stars have explicit conical product charts

Put $L=\mathrm{\mid }{\mathrm{lk}}_K\sigma \mathrm{\mid }$ and let $E=\{rq:0\le r<1,q\in L\}$ inside the Euclidean coordinate space on the vertices of the link, with the subspace topology. For an empty link both the cone by convention and $E=\{0\}$ are a single apex, so assume $L\mathrm{\ne }\mathrm{\varnothing }$. The continuous map $L\times [0,1)\to E$, $(q,r)\mapsto rq$, identifies exactly $L\times \{0\}$ with the apex: if $rq=r^{\mathrm{\prime }}{q}^{\mathrm{\prime }}$ with $r,r^{\mathrm{\prime }}>0$, coordinate sums give $r=r^{\mathrm{\prime }}$ and then $q=q^{\mathrm{\prime }}$. It therefore descends to a continuous bijection $\phi :c_{<1}(L)\to E$. For each $0<t<1$, the closed radial cone $(L\times [0,t])\mathrm{/}(L\times \{0\})$ is compact by [F2], and its continuous bijection onto $E_t=\{rq:0\le r\le t,q\in L\}$ is a homeomorphism because $E_t$ is Hausdorff. Away from the apex the inverse on the full open cone is continuous by $r={\sum }_v{z}_v$, $q=z\mathrm{/}r$, followed by the quotient projection. At the apex, the preimage of any quotient-open neighbourhood is an open set containing $L\times \{0\}$. Product neighbourhoods of its points, a finite cover of the compact link $L$, and the minimum of their positive radii give an $ϵ>0$ with $L\times [0,ϵ)$ in that preimage. The relatively open set $\{z\in E:{\sum }_v{z}_v<ϵ\}$ is consequently mapped by ${\phi }^{-1}$ into the given neighbourhood. Thus the inverse is continuous at the apex too, and $\phi$ is a homeomorphism of the full open cone.

For a point $x$ of the open star put $a={\sum }_{v\in \sigma }{x}_v>0$, set $p_v=x_v\mathrm{/}a$ for $v\in \sigma$ and $z_v=x_v$ for $v\mathrm{\notin }\sigma$. Then $p\in \mathrm{relint}\mathrm{\mid }\sigma \mathrm{\mid }$, $r:={\sum }_v{z}_v=1-a<1$, and the support of $x$ is $\sigma \cup \mathrm{supp}(z)$, so $\mathrm{supp}(z)$, if nonempty, is a simplex of the link. Conversely, given $p\in \mathrm{relint}\mathrm{\mid }\sigma \mathrm{\mid }$ and $z\in E$ with $r={\sum }_v{z}_v$, define a coordinate family by $x_v=(1-r)p_v$ on $\sigma$ and $x_v=z_v$ off $\sigma$; it is nonnegative with sum one, and its support is $\sigma \cup \mathrm{supp}(z)$, a simplex of $K$ because $\mathrm{supp}(z)$ is a simplex of the link. The two constructions are mutually inverse: from $x$ one recovers $a=1-r$ and $z$, hence $x$; from $(p,z)$ one recovers $p$ from the normalized $\sigma$-coordinates and $z$ from the outside coordinates. Both are continuous, the only division being by $a>0$ on the open star for the forward map, while the inverse is a polynomial expression in $(p,z)$ and the continuous function $r={\sum }_v{z}_v$. By step 1.1 the second factor is the cone $c_{<1}(L)$, and since $\mathrm{supp}(z)$ records exactly which link simplex the radial coordinate lies over, each simplex interior containing $\sigma$ is carried to the product of the corresponding link-simplex interior with $(0,1)$, while points with $z=0$ map to $\mathrm{relint}\mathrm{\mid }\sigma \mathrm{\mid }\times \{\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{x}\}$ and the simplex $\sigma$ itself to the apex. Thus the star is homeomorphic to the stated product.

The open star is open in $\mathrm{\mid }K\mathrm{\mid }$ because it is the finite intersection, over the vertices of $\sigma$, of the sets on which one barycentric coordinate is positive, each open in the weak topology by definition. The link is a subcomplex of the finite complex $K$, hence it has finitely many simplices and compact realization by [F2]. Finally, if a point $x$ of the star lay in $\mathrm{\mid }D\mathrm{\mid }$, its support would be a simplex $\rho$ of $D$ with $\sigma \subseteq \mathrm{supp}(x)=\rho$; the subcomplex $D$ is closed under taking faces, so $\sigma \in D$, contradicting $\sigma \mathrm{\notin }D$. Hence the star is disjoint from $\mathrm{\mid }D\mathrm{\mid }$.