Sub-Basis for Real Number Line

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Theorem

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


Then:

$\mathcal B := \left\{{\left({-\infty \,.\,.\, a}\right), \left({b \,.\,.\, \infty}\right): a, b \in \R}\right\}$ is a sub-basis for $\R$.


Proof

Let $\left({c \,.\,.\, d}\right)$ be an open real interval.

Then by definition:

$\left({c \,.\,.\, d}\right) = \left({-\infty \,.\,.\, d}\right) \cap \left({c \,.\,.\, \infty}\right)$

and so $\left({c \,.\,.\, d}\right)$ is the intersection of two elements of $\mathcal B$.


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