Sequence of Implications of Connectedness Properties

Theorem
Let $P_1$ and $P_2$ be connectedness 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:


 * Ultraconnected Space is Path-Connected


 * Arc-Connected Space is Path-Connected


 * Path-Connected Space is Connected


 * Irreducible Space is Connected