Sequence of Implications of Disconnectedness Properties

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:

Proof
The relevant justifications are listed as follows:


 * Discrete Space is $T_0$ and Discrete Space is Zero Dimensional.


 * Discrete Space is $T_1$ and Discrete Space is Scattered.


 * Zero Dimensional Space is T3, and by definition of a regular space as being both $T_3$ and $T_0$.


 * Zero Dimensional T0 Space is Totally Separated.


 * Discrete Space is Extremally Disconnected.


 * Extremally Disconnected Set is Totally Separated.


 * Extremally Disconnected Set is Totally Separated.


 * Totally Separated Space is Completely Hausdorff and Urysohn.


 * Totally Separated Space is Totally Disconnected.


 * Scattered T1 Space is Totally Disconnected.


 * Totally Disconnected Space is Totally Pathwise Disconnected.


 * Totally Disconnected Space is $T_1$.