In this paper, we prove the following statements: (1) There exists a pseudocompact star $\sigma$-compact Tychonoff space having a regular-closed subspace which is not star $\sigma$-compact. (2) Assuming $2^{\aleph_0}=2^{\aleph_1}$, there exists a star countable (hence star $\sigma$-compact) normal space having a regular-closed subspace which is not star $\sigma$-compact.