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:


Ultraconnected
$\Big\Downarrow$
Arc-Connected $\implies$ Path-Connected
$\Big\Downarrow$
Irreducible $\implies$ Connected


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

$\blacksquare$


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
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