x1 (metric space/set/class/structure) is a metric subspace/subset(/subclass) of x2 (metric space/set/class) which is closed therein. x1 is an closed subset of x2, where closedness is taken to be understood as being considered within x2 and under the metric shared by x1 and x2
In x2, x1 is composed exactly of members which are limit points of x1. Confer: sezrurkle.