I'm reading Regular Polytopes by H.S.M.Coxeter and have a question about simple-connectedness.
Coxeter says on p.1 (and similarly on p.9) that being simply-connected means that 'every simple closed curve drawn in the region can be shrunk to a point without leaving the region, i.e. there are no holes.' Then that such simply connected polyhedra satisfy Euler's formula, $N_0-N_1+N_2=2$.
My question regards solids which are composed of two polyhedra joined at a vertex, e.g.
Euler's formula fails for this solid, not surprisingly given that the formula applies to the original two tetrahedron separately, and so before being joined the total number of vertices minus edges plus faces equals four, so combining two vertices into one makes $V-E+F=3$. It seems to me that any simple closed curve drawn on the solid can be shrunk to a point however - either to the central joining point, or if necessary to one side of the solid. Am I wrong in thinking this, or are such solids just ruled out for some other reason (I notice that a solid like the above is mentioned by Lakatos in Proofs and Refutations p. 16 and David Richeson in Euler's Gem p.149, albeit the latter adds that 'it is debatable whether these figures should be classified as polyhedra').
Is there a simple way to rule out such solids? I notice that Coxeter gives as an alternate definition of simple-connectedness that 'every circuit of edges bounds a region' (p.9) which seems to me to be equivalent, but maybe rules out a solid such as the above?