Sub-basis for Uniformity on Real Number Space
Theorem
Let $\left({\R, \tau_d}\right)$ be the real number line under the Euclidean metric considered as a topological space.
Let $a, b \in \R$ such that $a < b$.
Let $S_{ab}$ be the set of subsets of $\R$ defined as:
- $S_{ab} = \left\{{\left({x, y}\right): x, y < b \text{ or } x, y > a}\right\}$
Then $S_{ab}$ is a basis for a uniformity $U$ which generates the usual topology on $\R$.
Note that $U$ is clearly not the usual metric uniformity.
In particular: Lots of assumed definitions here which might not be properly specified..
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Proof
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28: \ 8$
