Compactness Properties Preserved under Continuous Mapping/Mistake

Source Work

 * Part $\text{I}$: Basic Definitions
 * Section $3.$ Compactness
 * Invariance Properties
 * Invariance Properties

This mistake can be seen in the second edition (1978) as republished by Dover in 1995: ISBN 0-486-68735-X

Mistake

 * "To be precise, the properties of compactness, $\sigma$-compactness, countable compactness, sequential compactness, Lindelöf, and separability are preserved under continuous maps ... 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 $\left[{0 \,.\,.\, 1}\right]$ (which is compact), to $\R$ (which is not).

Also see

 * Compactness Properties Preserved under Continuous Surjection
 * Local Compactness is Preserved under Open Continuous Surjection
 * Countability Axioms Preserved under Open Continuous Surjection