Real Number Line less Zero is Disconnected Space
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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