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?
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.