Compactness Properties Preserved under Continuous Mapping/Mistake

Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):

Part $\text I$: Basic Definitions

Section $3$. Compactness

Invariance Properties

Mistake

To be precise, the properties of compactness, $\sigma$-compactness, countable compactness, sequential compactness, Lindelöf, and separability are preserved under continuous maps ... [Weak] local compactness, and first and second countability are preserved under open continuous maps, but not just under continuous maps ...

These statements are inaccurate.

In order for a mapping to preserve these properties, it also needs to be surjective.

As an illustrative example, consider the inclusion mapping from $\closedint 0 1$ (which is compact), to $\R$ (which is not).

Also see

• Compactness Properties Preserved under Continuous Surjection
• Weak Local Compactness is Preserved under Open Continuous Surjection
• Local Compactness is Preserved under Open Continuous Surjection
• Countability Axioms Preserved under Open Continuous Surjection

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: Invariance Properties