Theorem: Sequence of Implications of Disconnectedness Properties

Let $P_1$ and $P_2$ be disconnectedness properties.
Let:
:$P_1 \implies P_2$
mean:
:If a topological space $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | Regular||
| align="center" | $\impliedby$ ||
| align="center" | Zero Dimensional and $T_0$||
| align="center" | $\impliedby$ ||
| align="center" | Discrete Space ||
| align="center" | $\implies$ ||
| align="center" | $\implies$ ||
| align="center" | $\implies$ ||
| align="center" | Scattered and $T_1$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | Extremally Disconnected Hausdorff ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\implies$ ||
| align="center" | $\implies$ ||
| align="center" | Totally Separated ||
| align="center" | $\implies$ ||
| align="center" | Urysohn ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $T_1$ ||
| align="center" | $\impliedby$ ||
| align="center" | Totally Disconnected ||
| align="center" | $\impliedby$ ||
| align="center" | $\impliedby$ ||
| align="center" | $\impliedby$ ||
| align="center" | $\impliedby$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | Totally Pathwise Disconnected ||
|}
If there is no implication indicated, then there exists a counterexample demonstrating that no such implication exists.

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 $T_3$, and by definition of a regular space as being both $T_3$ and $T_0$
:Zero Dimensional $T_0$ Space is Totally Separated
:Discrete Space is Extremally Disconnected Hausdorff
:Extremally Disconnected Hausdorff Space is Totally Separated
:Totally Separated Space is Urysohn
:Totally Separated Space is Totally Disconnected
:Scattered $T_1$ Space is Totally Disconnected
:Totally Disconnected Space is Totally Pathwise Disconnected
:Totally Disconnected Space is $T_1$
{{qed|lemma}}
Then we have:
:Existence of Connected Non-$T_1$ Scattered Space
:Scattered Space is not necessarily $T_1$
:Zero Dimensional Space is not necessarily $T_0$
:Existence of Connected Space which is Totally Pathwise Disconnected
{{qed}}
{{finish}}