2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159736Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $π_1$-shape injective.12 pages, 5 figuresGeometric TopologyAlgebraic TopologyGroup Theory57N13, 57N65;19J25, 57P10, 57R67text