Is coordinated transformation which is not surjective (so it's inverse remove some region on spacetime) allowed in general relativity?
A coordinate chart does not need to cover the entirety of spacetime. So coordinate charts do not need to be surjective.
A coordinate transformation ( I.e. the map relation to coordinate charts) only needs to be valid on the part of the spacetime covered by both charts. This means that a coordinate transformation does not have to be valid on the entire domain of the original coordinate chart, nor does its image have to cover the entirety of the domain of the new coordinate chart, i.e. it does not have to be surjective.