# Clarifying the definition of Boundedness in Uniform Spaces

This excerpt from a journal paper says if X is a uniform space and $$A$$ is a subset of $$X$$, then $$A$$ is said to be bounded if for each entourage $$V$$, "there exists a finite set $$F$$ and a positive integer $$m$$ such that $$A\subseteq V^m[F]$$". My question is, what does $$V^m[F]$$ mean?

The $$V^m$$ part is clear enough; you just compose $$V$$ with itself $$m$$ times. But I haven't encountered a set in brackets after an entourage. If it said $$V^m[x]$$ where $$x\in X$$, that would make complete sense. How are the two notations related? If $$F=\{x_1,...,x_n\}$$, then does $$V^m[F]=\cup_{i=1}^n V^m[x_i]$$ or something?

If $$R$$ is a relation on $$X$$ (so $$R \subseteq X \times X$$) then $$R[F]$$ is just the "functional image" for a subset $$F$$ of $$X$$:
$$R[F]=\{y \in X: \exists x \in F: (x,y) \in R\}$$
So yes, this does commute with unions as you suggested. This is the same definition in essence as $$f[F]$$ for a function $$f: X \to Y$$ and $$F \subseteq X$$; note that a function is just a special case of a relation.