Countably Compact Space is Weakly Countably Compact
Jump to navigation
Jump to search
Theorem
Every countably compact space is a weakly countably compact space.
Proof
Let $T = \struct {S, \tau}$ be a countably compact space.
By definition:
- $T$ is weakly countably compact if and only if every infinite subset of $S$ has a limit point in $S$.
By definition of countably compact space:
- every infinite subset of $S$ has an $\omega$-accumulation point in $S$.
By definition, an $\omega$-accumulation point is a limit point.
Hence the result.
$\blacksquare$
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