Real Number Space is Paracompact

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Theorem

Let $\left({\R, \tau_d}\right)$ be the real number line considered as a topological space under the usual (Euclidean) topology.


Then $\left({\R, \tau_d}\right)$ is paracompact.


Proof

Let $\mathcal C$ be an open cover for $\R$.

Then $\mathcal C$ covers each of the closed real intervals $\left[{n \,.\,.\, n + 1}\right]$ for all $n \in \Z$.

By the Heine-Borel Theorem, each of $\left[{n \,.\,.\, n + 1}\right]$ is compact.

So, for each of these intervals $\left[{n \,.\,.\, n + 1}\right]$, it follows that $\mathcal C$ can be reduced to a sequence $\left \langle {G_i^{(n)}}\right \rangle$ of finite subcovers.

Then each of $G_i^{(n)} \cap \left({n - 1 \,.\,.\, n + 2}\right)$ forms a refinement of $\mathcal C$ which is locally finite.

Hence, by definition, $\R$ is paracompact.

$\blacksquare$


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