Is NPSPACE also closed under polynomial-time reduction and under log-space reduction? The complexity classes P, NP, and PSPACE are closed under polynomial-time reduction.

The complexity classes L, NL, P, NP and PSPACE are closed under log-space reduction.

I wonder if NPSPACE is also closed under polynomial-time reduction and/or under log-space reduction.
Savitch's theorem implies that NPSPACE is equal to PSPACE, and so it is closed under polynomial time reductions.