Sequence of Implications of Disconnectedness Properties
Jump to navigation
Jump to search
Theorem
Let $P_1$ and $P_2$ be disconnectedness properties and let:
- $P_1 \implies P_2$
mean:
- If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
Regular | $\impliedby$ | Zero Dimensional and $T_0$ | $\impliedby$ | Discrete Space | $\implies$ | $\implies$ | $\implies$ | Scattered and $T_1$ | |||||||||
$\Big\Downarrow$ | $\Big\Downarrow$ | $\Big\Downarrow$ | |||||||||||||||
$\Big\Downarrow$ | Extremally Disconnected | $\Big\Downarrow$ | |||||||||||||||
$\Big\Downarrow$ | $\Big\Downarrow$ | $\Big\Downarrow$ | |||||||||||||||
$\implies$ | $\implies$ | Totally Separated | $\implies$ | Urysohn | $\Big\Downarrow$ | ||||||||||||
$\Big\Downarrow$ | $\Big\Downarrow$ | ||||||||||||||||
$T_1$ | $\impliedby$ | Totally Disconnected | $\impliedby$ | $\impliedby$ | $\impliedby$ | $\impliedby$ | |||||||||||
$\Big\Downarrow$ | |||||||||||||||||
Totally Pathwise Disconnected |
Proof
The relevant justifications are listed as follows:
- Zero Dimensional Space is T3, and by definition of a regular space as being both $T_3$ and $T_0$.
$\blacksquare$
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