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 ... [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 $\left[{0 \,.\,.\, 1}\right]$ (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