# Compactness Properties Preserved under Continuous Mapping/Mistake

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## 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

- Section $3$. Compactness

## 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