The results concern clopen sets in products of topological spaces. It is shown that a clopen subset of the product of two separable metrizable (or locally compact) spaces is not always a union of clopen boxes. It is also proved that any clopen subset of the product of two spaces, one of which is compact, can always be represented as a union of clopen boxes.
Let be a continuum and a positive integer. Let be the hyperspace of all nonempty closed subsets of with at most components, endowed with the Hausdorff metric. For compact subset of , define the hyperspace . In this paper, we consider the hyperspace , which can be a tool to study the space . We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.