An edge path in a simplicial complex is a finite sequence of vertices of a simplicial complex such that is a simplex for each consecutive pair. Repeated consecutive vertices are allowed. Its geometric realization is a continuous path formed by traversing the corresponding line segments, with constant pieces for repetitions.
A connected geometric realization has an edge path between any two vertices. For a fixed vertex, the reachable vertices form a union of whole simplexes: all vertices of a simplex are pairwise joined by edges. The reachable and unreachable geometric parts therefore intersect each simplex in either the whole simplex or the empty set. Both parts are closed in the realization topology and are complements, so both are open. Connectedness forces the unreachable part to be empty. Conversely, edge paths between vertices and straight segments inside simplexes join arbitrary points, so the realization is path-connected.
New to topics? Read the docs here!