Sub-Basis for Real Number Line

Theorem

Let the real number line $\R$ be considered as a topology under the usual (Euclidean) topology.

Then:

$\BB := \set {\openint \gets a, \openint b \to: a, b \in \R}$ is a sub-basis for $\R$.

Proof

Let $\openint c d$ be an open real interval.

Then by definition:

$\openint c d = \openint \gets d \cap \openint c \to$

and so $\openint c d$ is the intersection of two elements of $\BB$.

From Open Sets in Real Number Line, any open set of $\R$ is the union of countably many open real intervals.

The result follows by definition of sub-basis.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.3$: Sub-bases and weak topologies: Example $3.3.1$