On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds

We show that if an Alexandrov space $X$ has an Alexandrov subspace $\bar Ω$ of the same dimension disjoint from the boundary of $X$, then the topological boundary of $\bar Ω$ coincides with its Alexandrov boundary. Similarly, if a noncollapsed RCD(K,N) space $X$ has a noncollapsed RCD(K,N) subspace $\bar Ω$ disjoint from boundary of $X$ and with mild boundary condition, then the topological boundary of $\bar Ω$ coincides with its De Philippis-Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.