We study a class of spaces whose importance in homotopy theory was first highlighted by work of Dold in the 1960s, and that we accordingly call Dold spaces. These are the spaces that possess a partition of unity supported in sets that are contractible to a point within the ambient space. Dold spaces form a broader class than spaces homotopy equivalent to CW complexes, but share the feature that a number of well known weak equivalences are genuine ones if Dold spaces are involved. In this paper we give a first systematic investigation of Dold spaces. After listing their elementary properties, we study homotopy pullbacks involving Dold spaces and simplicial objects in the category of Dold spaces. In particular, we show that the homotopy colimit of a diagram of Dold spaces is a Dold space and that the topological realization functor preserves fibration sequences if the base is a path-connected Dold space in each dimension. It follows that the loop space functor commutes with realization up to homotopy for Dold spaces. Finally, we give simple conditions which assure that free algebras over a topological operad are Dold spaces.
"Dold spaces in homotopy theory." Algebr. Geom. Topol. 9 (3) 1585 - 1622, 2009. https://doi.org/10.2140/agt.2009.9.1585