 In other words, a function which is an isomorphism on formula_20 only implies that any elements of formula_21 that are homotopic in "X" are "abstractly" homotopic in "A" – the homotopy in "A" may be unrelated to the homotopy in "X" – while being "n"-connected (so also onto formula_22) means that (up to dimension "n" − 1) homotopies in "X" can be pushed into homotopies in "A".This gives a more concrete explanation for the utility of the definition of "n"-connectedness