We apply the theory of directed topology developed by Grandis [Directed homotopy theory, I. The fundamental category, Cah. Topol. Géom. Différ. Catég. 44 (2003) 281–316; Directed Algebraic Topology: Models of Non-Reversible Worlds, New Mathematical Monographs, Vol. 13 (Cambridge University Press, Cambridge, 2009)] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [A. L. Dougherty, H. Park and D. N. Yetter, On 2-dimensional Dikjgraaf-Witten theory with defects, J. Knot Theory Ramifications 25(5) (2016) 1650021, doi:10.1142/S0218216516500218 ; I. J. Lee and D. N. Yetter, Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds, J. Knot Theory Ramifications 26(5) (2017) 1750026, doi:10.1142/S0218216517500262 ], which we extend to a similar construction of a Dijkgraaf–Witten type TQFT in the case where the defects (lower-dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orientation convention.