Obravnavamo obnašanje vlaknenj pri tvorejnju preslikavnih prostorov, t.j. pri danem vlaknenju ▫$p \colon E \to B$▫ poskušamo ugotoviti za katere prostore ▫$X$▫ je tudi inducirana preslikava ▫$p_\ast ...\colon E^X \to B^X$▫ vlaknenje. Če je ▫$X$▫ lokalno kompakten, nam eksponentni zakon zlahka da pozitiven odgovor. Zato se tukaj ukvarjamo z bolj splošnimi prostori in pokažemo, da je ohranitev vlaknenj povezana s homotopskimi lastnostmi prostora ▫$X$▫. Na primer, če je ▫$p$▫ Doldovo vlaknenje in če ▫$X$▫ lahko deformacijsko retraktiramo na kompaktno generiran prostor, potem je tudi inducirana preslikava ▫$p_\ast$▫ Doldovo vlaknenje. Podobno velja tudi za Hurewiczeva vlaknenja z enoličnim dvigom poti in za krovne prostore.
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