Basis for Euclidean Topology on Real Number Line
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Theorem
Let $\R$ be the set of real numbers.
Let $\BB$ be the set of subsets of $\R$ defined as:
- $\BB = \set {\openint a b: a, b \in \R}$
That is, $\BB$ is the set of all open real intervals of $\R$:
- $\openint a b := \set {x \in \R: a < x < b}$
Then $\BB$ forms a basis for the Euclidean topology on $\R$.
Proof
From Real Number Line is Metric Space, one can define an open interval on the set of real numbers in terms of an $\epsilon$-neighborhood.
From Open Real Interval is Open Ball, an open interval $\openint a b$ is the open $\epsilon$-ball $\map {B_\epsilon} \alpha$.
Then from Metric Induces Topology we have that:
- $\BB = \set {\openint a b: a, b \in \R}$
forms a topology on $\R$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology