Discrete Space is Extremally Disconnected
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Then $T$ is extremally disconnected.
Proof
First we note that as Discrete Space satisfies all Separation Properties, $T$ is a $T_2$ (Hausdorff) space.
Then from Interior Equals Closure of Subset of Discrete Space, it follows directly that the closure of every open set of $T$ is open.
Hence by definition $T$ is extremally disconnected.
$\blacksquare$
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