For \(X\) a nonempty topological space, \(k\) a positive integer, the author considers \(\text{Sub}(X,k)\) the set of nonempty subsets of \(X\) having cardinality \(\leq k\), topologized such that for \(i=1,\dots,k\) the configuration space \(C(X,i)\) with its standard topology will be a subspace of \(\text{Sub} (X,k)\). First, the author establishes some general topological properties and some homotopy properties for \(\text{Sub}(X,k)\). The \(\text{Sub}(\cdot,k)\) are homotopy functors and their properties are studied.
The first main result is that if \(X\) is a nonempty path-connected Hausdorff space, then for each \(k\geq 1\) and \(n\geq 0\) the map \(\pi_n(\text{Sub}(X,k)) \to\pi_n( \text{Sub} (X,2k+1))\) induced by the inclusion is the 0-map.
In contrast with this (in the direction of nontriviality) the second main result is that if \(X\) is a nonempty closed manifold of dimension \(\geq 2\), then \(\text{Sub} (X,k)\) is homologically nontrivial for all \(k\geq 1\).