  We study the Penrose inequality and its rigidity for metrics with singular
sets. Our result could be viewed as a complement of Theorem 1.1 of Lu and Miao
(J. Funct. Anal. 281, 2021) and Theorem 1.2 of Shi, Wang and Yu (Math. Z. 291,
2019), in which they assume the singular set is a hypersurface and assume an
additional condition on the mean curvature. As a complement, this paper study
the case of singular set of dimensional less than n-1, without any additional
conditions.
