Sequence of Implications of Disconnectedness Properties

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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:


$\blacksquare$


Sources