  We introduce partial loop-erasing operators. We show that by applying a
refinement sequence of partial loop-erasing operators to a finite Markov chain,
we get a process equivalent to the chronological loop-erased Markov chain. As
an application, we construct loop-erased random paths on bounded domains of
resistance spaces as the weak limit of the loop erasure of the Markov chains on
a sequence of finite sets approximating the space, and the limit is independent
of the approximating sequences. The random paths we constructed are simple
paths almost surely, and they can be viewed as the loop-erasure of the paths of
the diffusion process. Finally, we show that the scaling limit of the
loop-erased random walks on the Sierpi\'nski carpet graphs exists, and is
equivalent to the loop-erased random paths on the Sierpi\'nksi carpet.
