Sub-basis for Uniformity on Real Number Line

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $a, b \in \R$ such that $a < b$.

Let $S_{a b}$ be the set of subsets of $\R$ defined as:

$S_{a b} = \set {\tuple {x, y}: x, y < b \text{ or } x, y > a}$

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.

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Proof

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Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $8$