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

 * "... The properties of compactness, $\sigma$-compactness, countable compactness, sequential compactness, Lindelöf, and separability are preserved under continuous maps ..."


 * "Local compactness ... [is] preserved under open continuous maps, but not just under continuous maps ..."

These statements are inaccurate.

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

Consider the inclusion mapping from $\left[{0..1}\right]$ (which is compact), to $\R$ (which is not).

Also see

 * Compactness Properties Preserved under Continuous Surjections
 * Local Compactness Preserved under Open Continuous Surjections