Union of Exteriors contains Exterior of Intersection/Mistake

Source Work

 * Part $\text{I}$: Basic Definitions
 * Section $1.$ General Introduction
 * Closures and Interiors
 * Closures and Interiors

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

Mistake

 * "The exterior of the union of sets is always contained in the intersection of the exteriors, and similarly, the exterior of the intersection is contained in the union of the exteriors; equality holds only for finite unions and intersections."

Part of the above statement is true. See:
 * Intersection of Exteriors contains Exterior of Union
 * Exterior of Finite Union equals Intersection of Exteriors

However, the statement:
 * " ... the exterior of the intersection is contained in the union of the exteriors ..."

is shown to be false by Exterior of Intersection contains Union of Exteriors.

The statement:
 * "... equality holds only for finite unions and intersections ..." is also shown to be untrue, as follows.

Consider where $T = \R$, the real number line.

Let $H = \left({0 . . 2}\right) \cup \left({3 . . 4}\right), K = \left({1 . . 3}\right)$.

For these sets:

We see that $3 \in \left({H \cap K}\right)^e$, but $3 \notin H^e \cup K^e$.

So $H^e \cup K^e \subset \left({H \cap K}\right)^e$ but $H^e \cup K^e \ne \left({H \cap K}\right)^e$.

Acknowledgements
This error was confirmed by HappyJoe on the (inconveniently intermittent) mathhelpforum on 17th April 2011.