Real Number Space is not Countably 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 not countably compact.


Proof

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

$\mathcal C = \left\{{\left({n \,.\,.\, n + 2}\right): n \in \Z}\right\}$

Then $\mathcal C$ is an open cover of $\R$ which is countable.

However, there is no finite subcover for $\R$ of $\mathcal C$.

Hence the result, by definition of countably compact.

$\blacksquare$


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