Example of a connected, finite simplicial complex that is not a closed surface I need to find an example of a connected finite SC that is not a closed combinatorial surface, but satisfying 1. contains only 0 1 2 simplices 2. every 1-simplex is a face of precisely two 2-simplices 3. every point of |K| lies in a 2 simplex. 
The first example I came out with is tetrahedron. But I think it is a closed surface. Should I delete some point to make it not closed? 
 A: Take a triangulation of the sphere with a lot of triangles in it. Then glue two of the vertices together in such a way that the result is still a simplicial complex. This can't be done for the tetrahedron since every pair of vertices share an edge, but is possible for larger triangulations.
A: Take two tetrahedrons and join them on a vertex. The link of the join is then a union of two disjoint simplicial circles, or alternatively, any neighbourhood of the join can be disconnected by removing the point of the join, whereas there exists no open set in $\mathbb{R}^2$ that can be disconnected by removing a single point.
More specifically, consider the topological realisation of the following abstract simplicial complex $(V, \Sigma)$
$
V = \{\ 0, 1, 2, 3, 11, 12, 13\ \}
$
$
\Sigma =  \wp(\{\ 0, 1, 2, 3\ \}) \\ \ \ \ \ \ \ \ \ \ \ \ \ \cup \ \wp(\{\ 11, 12, 13\ \}) \\ \ \ \ \ \ \ \ \ \ \ \ \  \cup \ \{\{\ 0, 11\ \},\{\ 0, 12\ \},\{\ 0, 13\ \},\{\ 0, 11, 12\ \},\{\ 0, 11, 13\ \},\{\ 0, 12, 13\ \}\}
$
