Definition:Discrete Uniformity
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Definition
Let $S$ be a set.
The discrete uniformity on $S$ is the uniformity $\UU$ defined as:
- $\UU := \set {u \subseteq S \times S: \Delta_S \subseteq u}$
that is, all subsets of the cartesian product on $S$ which contain the diagonal relation on $S$.
Hence from Relation Contains Diagonal Relation iff Reflexive it can be considered as the set of all reflexive relations on $S$.
Also see
- Results about discrete uniformities can be found here.
Linguistic Note
Be careful with the word discrete.
A common homophone horror is to use the word discreet instead.
However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $11$