Basis for Element of Real Number Line

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\R$ be the real number line considered as a metric space under the usual metric.

Let $a \in \R$ be a point in $\R$.


Then the set of all open intervals containing $a$ is a basis for the neighborhood system of $a$.


Proof

Let $N$ be a neighborhood of $a$ in $M$.

Then by definition:

$\exists \epsilon \in \R_{>0}: B_\epsilon \left({a}\right) \subseteq N$

where $B_\epsilon \left({a}\right)$ is the open $\epsilon$-ball at $a$.

The result follows from Open Ball in Real Number Line is Open Interval.

$\blacksquare$


Sources