On small homotopies of loops
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Two 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 figures
12 pages, 5 figures