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1-1 correspondence abstract spaces axioms bicompact boundary called Cantor set circle closed and bounded closed sets collection of numbers compact complement connected contains no point continuum countable deﬁned denote disk distance function elements Euclidean plane examples of topological Exercise exterior False Statement ﬁgure ﬁnd ﬁrst function space geometry graph homeomorphism horned sphere integer interior intersect intuitive Jordan curve theorem Klein bottle least upper bound lies limit point mathematics metric space Moebius band mutually separated open set open subset ordered pairs point set topology polygon positive number projective plane prove pulling and stretching real numbers rectangle satisfy Property Schoenﬂies theorem section Some examples set of points sets is closed shelling theorem shown in Figure sides simple closed curve square Suppose surface takes topological characterization topological spaces topologically equivalent triangle undeﬁned terms unsolved problems vertical line x-axis Y+ YZ