  We initiate the study of pointed approximative absolute neighborhood
retracts. Our motivation is to generate examples of C*-algebras that behave in
unexpected ways with respect to weak semiprojectivity. We consider both weak
semiprojectivity (WSP) and weak semiprojectivity with respect to the class of
unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different
properties.
  One example shows a C*-algebra can fail to be WSP while its unitization is
WSP. Another example shows WSP1 is not closed under direct sums.
