Given a compact metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighbourhood retract of dimension at most 1. This confirms a conjecture of Blackadar. Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As applications of our results, we prove two theorems about the structure of semiprojective commutative C*-algebras. Letting A be a commutative C*-algebra, we show firstly: If I is an ideal of A and A/I is finite-dimensional, then A is semiprojective if and only if I is; and secondly: A is semiprojective if and only if M2(A) is. This answers two questions about semiprojective C*-algebras in the commutative case.
London Mathematical Society. Proceedings, 2012, Vol 105, Issue 5, p. 1021-1046