Totally Disconnected Space is T1
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected.
Then $T$ is a $T_1$ (Fréchet) space.
Proof
Let $T = \struct {S, \tau}$ be totally disconnected.
Then as its components are singletons, it follows that each of its points is closed.
From Equivalence of Definitions of $T_1$ Space, it follows that $T$ is a $T_1$ (Fréchet) space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness