Cantor Space is Meager in Closed Unit Interval

Theorem

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.

Then $T$ is meager in $\closedint 0 1$.

Proof

From Cantor Space is Nowhere Dense, $T$ is nowhere dense in $\closedint 0 1$.

So, trivially, $\CC$ is the union of nowhere dense subsets of $\closedint 0 1$.

Hence the result from definition of meager.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set: $4$