The aim of this paper is to explore some properties of quasiuniform multifunction spaces. Various kinds of completeness of the quasiuniform multifunction space (YmX, UmX) are characterized in terms of suitable properties of the range space (Y, U). We also discuss the local compactness of quasiuniform multifunction spaces. By using the notion of small-set symmetry, the classic result of Hunsaker and Naimpally is extended to the quasiuniform setting.

A general procedure is introduced for identifying categories of Cauchy spaces which have completions which possess a certain convergence property P, and for constructing completion functors on such categories. These results are applied to obtain a characterization of Cauchy spaces which allow T3 completions, and to construct T3 completions for categories of Cauchy groups and Cauchy lattices.

A uniform space X is said to be uniformly locally connected if given any entourage U there exists an entourage V ⊂ U such that V[x] is connected for each x ∈ X. It is said to have property S if given any entourage U, X can be written as a finite union of connected sets each of which is U-small.

Based on these two uniform connection properties, another proof is given of the following well known result in the theory of locally connected spaces: The Stone-Čech compactification βX is locally connected if and only if X is locally connected and pseudocompact.

Relations between uniform paracompactness and uniform para-Lindelöfness of a uniform space and its uniform weight are established.

We prove that a topological space X has a locally connected regular T1, extension if and only if X is the underlying topological space of a nearness space Y which is concrete, regular and uniformly locally uniformly connected.

A theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.