Euclidean metric frames which are not spatial

In this paper, we employ the concept of a ζ ζ -ideal J J and introduce a quotient frame M(X,J) M ( X , J ) , where X is a topological space. We prove that for a metric space X , this frame is metrizable. Moreover, a necessary and sufficient condition for spatiality of frame M(X,J) M ( X , J ) will be given. Finally, we apply this general case to the metric space Rⁿ R n and construct frames which are metrizable but not spatial. More precisely, these frames are derived from the metric space Rⁿ R n which are actually the quotients of frame O(Rⁿ) O ( R n ) , the set of open subsets of Rⁿ R n . The technique of constructing these frames, which are denoted by Mᵢ(X, S) M i ( X , S ) , $$i=0,1,2$$ i = 0 , 1 , 2 , intuitively, is a process in which a given subset of points are removed.