Real Number Line is not Countably Compact

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.


Then $\struct {\R, \tau_d}$ is not countably compact.


Proof

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

$\CC = \set {\openint n {n + 2}: n \in \Z}$

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

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

Hence the result, by definition of countably compact.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $4$
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