Sequence of Implications of Connectedness Properties
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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:
$\blacksquare$
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