  We present a detailed low temperature scanning tunneling microscopy study of
the commensurate charge density wave (CDW) in 1$T$-TiSe$_2$ in the presence of
single atom defects. We find no significant modification of the CDW lattice in
single crystals with native defects concentrations where some bulk probes
already measure substantial reductions in the CDW phase transition signature.
Systematic analysis of STM micrographs combined with density functional theory
modelling of atomic defect patterns indicate that the observed CDW modulation
lies in the Se surface layer. The defect patterns clearly show there are no
2$H$-polytype inclusions in the CDW phase, as previously found at room
temperature [Titov A.N. et al, Phys. Sol. State 53, 1073 (2011). They further
provide an alternative explanation for the chiral Friedel oscillations recently
reported in this compound [J. Ishioka et al., Phys. Rev. B 84, 245125, (2011)].
