Real Number Space is Sigma-Compact

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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 $\sigma$- compact.


Proof

We have that a Real Number Space satisfies all Separation Axioms.

Specifically, $\left({\R, \tau_d}\right)$ is a Hausdorff space.

Consider $\mathcal C$ the set of subsets of $\R$ defined as:

$\mathcal C = \left\{{\left[{n \,.\,.\, n + 1}\right]: n \in \Z}\right\}$

where $\left[{n \,.\,.\, n + 1}\right]$ is the closed real interval between successive integers.

By the Heine-Borel Theorem, each element of $\mathcal C$ is compact.

$\mathcal C$ itself is countable, as there is a (trivial) one-to-one-correspondence between $\mathcal C$ and $\Z$.

Every element of $\R$ is contained in at least one of the elements of $\mathcal C$.

Thus $\R$ is the union of $\mathcal C$.

Hence, by definition, $\R$ is $\sigma$-compact.

$\blacksquare$


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