Sequence of Implications of Global Compactness Properties
Jump to navigation
Jump to search
Theorem
Let $P_1$ and $P_2$ be compactness 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:
Sequentially Compact | |||||||||
$\Big\Downarrow$ | |||||||||
Compact | $\implies$ | Countably Compact | $\implies$ | Pseudocompact | |||||
$\Big\Downarrow$ | $\Big\Downarrow$ | ||||||||
$\sigma$-Compact | Weakly Countably Compact | ||||||||
$\Big\Downarrow$ | |||||||||
Lindelöf Space |
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 $3$: Compactness: Global Compactness Properties