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'uniform space'
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| Title of object: |
uniform space |
| Canonical Name: |
UniformSpace |
| Type: |
Definition |
| Created on: |
2002-06-10 06:20:39.98387-04 |
| Modified on: |
2002-11-03 12:10:19.731848-05 |
| Classification: |
msc:54E15 |
| Defines: |
uniform structure, entourage |
Revision comment (for changes between this and next version):
| Changes for correction #1587 ('term 'uniformity' '). |
Preamble:
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Content:
A \emph{uniform structure} on a set $X$ is a non empty set $\mathcal{U}$ of subsets of $X \times X$ which satisfies the following axioms:
\begin{enumerate}
\item Every subset of $X\times X$ which contains a set of $\mathcal{U}$ belongs to $\mathcal{U}$
\item Every finite intersection of sets of $\mathcal{U}$ belongs to $\mathcal{U}$
\item Every set is the graph of a reflexive relation (i.e. contains the diagonal)
\item If $V$ belongs to $\mathcal{U}$, then $V' = \{(y,x): (x,y) \in V\}$ belongs to $\mathcal{U}$.
\item If $V$ belongs to $\mathcal{U}$, then exists $V'$ in $\mathcal{U}$ such that, whenever $(x,y),(y,z) \in V'$, then $(x,z) \in V$.
\end{enumerate}
The sets of $\mathcal{U}$ are called \emph{entourages}. The set $X$, together with the uniform structure $\mathcal{U}$, is called a \emph{uniform space}.
Every uniform space can be considered a topological space with a natural topology induced by uniform structure. The uniformity, however, provides in general a richer structure, which formalize the concept of relative closeness: in a uniform space we can say that $x$ is close to $y$ as $z$ is to $w$, which makes no sense in a topological space. It follows that uniform spaces are the most natural environment for uniformly continuous functions and Cauchy sequences, in which these concepts are naturally envolved.
Examples of uniform spaces are metric spaces and topological groups. |
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