For a path-connected space , a well-known theorem of Segal, May and Milgram asserts that the configuration space of finite points in with labels in is weakly homotopy equivalent to . In this paper, we introduce a space of intervals suitably topologized in with labels in a space and show that it is weakly homotopy equivalent to without the assumption on path-connectivity.
"The space of intervals in a Euclidean space." Algebr. Geom. Topol. 5 (4) 1555 - 1572, 2005. https://doi.org/10.2140/agt.2005.5.1555