Real Number Line is not Topological Continuum

Theorem

The real number line is not a continuum in the topological sense.

Proof

Recall the definition of continuum:

A topological space $T$ is a continuum if and only if $T$ is both compact and connected.

However, we have the result Real Number Line is not Compact.

Hence the result.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuum: 2. (plural continua)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continuum: 2. (plural continua)