Real Number Line less Zero is Disconnected Space

Theorem

Let $S := \R \setminus \set 0$ be the real number line with $0$ excluded.

Let $\struct {S, \tau_d}$ be $S$ with the usual (Euclidean) topology.

Then $\struct {S, \tau_d}$ is disconnected.

Proof

We note that:

$S = \openint \gets 0 \cup \openint 0 \to$

$\openint \gets 0 \cap \openint 0 \to = \O$

Hence we have partitioned $S$ into $2$ disjoint open sets whose union is $S$.

Hence by definition $S$ is not connected.

Hence the result by definition of disconnected space.

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): connected space