Real Number Line satisfies all Separation Axioms
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Theorem
Let $\struct {\R, \tau_d}$ be the the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ fulfils all separation axioms:
- $\struct {\R, \tau_d}$ is a $T_0$ (Kolmogorov) space
- $\struct {\R, \tau_d}$ is a $T_1$ (Fréchet) space
- $\struct {\R, \tau_d}$ is a $T_2$ (Hausdorff) space
- $\struct {\R, \tau_d}$ is a semiregular space
- $\struct {\R, \tau_d}$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space
- $\struct {\R, \tau_d}$ is a $T_3$ space
- $\struct {\R, \tau_d}$ is a regular space
- $\struct {\R, \tau_d}$ is an Urysohn space
- $\struct {\R, \tau_d}$ is a $T_{3 \frac 1 2}$ space
- $\struct {\R, \tau_d}$ is a Tychonoff (completely regular) space
- $\struct {\R, \tau_d}$ is a $T_4$ space
- $\struct {\R, \tau_d}$ is a normal space
- $\struct {\R, \tau_d}$ is a $T_5$ space
- $\struct {\R, \tau_d}$ is a completely normal space
- $\struct {\R, \tau_d}$ is a perfectly $T_4$ space
- $\struct {\R, \tau_d}$ is a perfectly normal space
Proof
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $1$