= Edge path
{title2=$(v_0,v_1,\ldots,v_n)$}
An edge path in a <simplicial complex> is a finite sequence of <vertices of a simplicial complex> such that $\{v_i,v_{i+1}\}$ 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>.
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