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0 1 If one has to prove that $R^n - {0}$ is not $(n-1)$ -connected, is it necessary to prove formally that there exists a non contractible $(n-1)$ -sphere or can that simply be stated. If one must formally prove this, how may that be done? Can one, for example, consider the intersection of the $(n-1)$ -sphere and $R^n$ with an arbitrary 2D plane and then say (for the sake of contradiction by the definition of 1-connectedness) that the given sphere is contractible if and only if the circle that corresponds to the intersection of that sphere is contractible in the subspace topology defined by the intersection of the plane and $R^n$ ? general-topology proof-verification geometric-topology share | cite | improve this...