I am trying to verify the following claims about closed surfaces from the book Basic Topology by M.A. Armstrong.
If h: |K|$\to$ S is a triangulation of a closed surface we are going to show that the simplicial complex K must be a combinatorial surface. Show that:
i) K cannot have dimension 1.
ii) K does not contain a simplex with dimension greater than 2.
iii) every edge in K lies in exactly two triangles.
iv) each vertex lies in at least three triangles whose union forms a cone with apex v.
I started with i). Suppose that the claim is false. Then there is a complex K such that dim(K)=1. This implies that no simplex in K may have dimension greater than 1. This means that all simplexes in K are either single vertices or straight line-segments between two vertices. Moreover we can guarantee the existence of at least one simplex with dimension one.
Since S is a closed surface it is compact, connected and hausdorff.
I also know that each two vertices of K can be joined by an edge path. That is, they can be joined by a sequence of 1-simplexes in K.
In order to produce a contradiction I have to find a 2-simplex contained in K.
So take a 1-simplex $\sigma$ in K with vertices v,w. For any vertex z in K there are edge paths joining v and w to z. I now have to prove that for some z that is neither v nor w that these edge paths consist only of a single edge.
The problem is that there seems to be no way to show that.