## A characterization of those subsets of metric separable space which are homomorphic with subsets of the linear continuum |

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1—1 correspondence Annalen class of sets closed arc component closed intervals complement conditions of lemma consequence of 1.1 consequence of lemma consequence of theorem contains a limit corres countable set definition of neighborhood denoted by U(q dense perfect set dense set element q elements of Q euclidean space exists an interval finite number following Theorem half-open arc Hence homeomorphic invariant interval complementary limit point linear continuum linear intervals linear set Math Necessary and sufficient need consider neighborhood U(q neighborhoods in Q non-overlapping open arc open interval open sets p(ZP ponent positive number positive rational values primary neighborhood Proof q(ZQ represent point components represent the components represents a point represents an arc right end-point satisfying the conditions separable metric space set of point set U(q sets A(T space be homeomorphic space Q subsets of U(q sufficient conditions terval theorem VIII totally disconnected sets U(qt