Existence of Compact Space which is not Sequentially Compact
Jump to navigation
Jump to search
Theorem
There exists at least one example of a compact topological space which is not also a sequentially compact space.
Proof
Let $T = \mathbb I^{\mathbb I}$ be the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.
From Uncountable Cartesian Product of Closed Unit Interval is Compact, $T$ is a compact space.
From Uncountable Cartesian Product of Closed Unit Interval is not Sequentially Compact, $T$ is not a sequentially compact space.
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