English Dictionary

First, let us define its 1-dimensional analog, that is, a topological graph. A graph 𝛥 is a 1-dimensional stratified topological space with finitely many 0-strata (vertices) and finitely many 1-strata (edges). […] A graph such that any vertex belongs to at least two half-edges we call an s-graph. Clearly the boundary ∂𝛺 of a surface 𝛺 with marked points is an s-graph. A morphism of graphs 𝜑:𝛥'→𝛥 is a continuous epimorphic map of graphs compatible with the stratification; i.e., the restriction of 𝜑 to any open 1-stratum (interior of an edge) of 𝛥' is a local (therefore, global) homeomorphism with appropriate open 1-stratum of 𝛥.