On small homotopies of loops

Two natural questions are answered in the negative:• space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?” dding arcs to a space cause an essential curve to become nulhomotopic?” nswer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π 1 -shape injective.