# Union of Exteriors contains Exterior of Intersection/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 $1$. General Introduction
- Closures and Interiors

- Section $1$. General Introduction

## 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.*

## Correction

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 $H = \openint 0 2 \cup \openint 3 4, K = \openint 1 3$.

For these sets:

\(\ds \paren {H \cap K}^e\) | \(=\) | \(\ds \openint {-\infty} 1 \cup \openint 2 \infty\) | ||||||||||||

\(\ds H^e \cup K^e\) | \(=\) | \(\ds \openint {-\infty} 1 \cup \openint 2 3 \cup \openint 3 \infty\) |

We see that $3 \in \paren {H \cap K}^e$, but $3 \notin H^e \cup K^e$.

So:

- $H^e \cup K^e \subset \paren {H \cap K}^e$

but:

- $H^e \cup K^e \ne \paren {H \cap K}^e$

## Acknowledgements

This error was confirmed by HappyJoe on mathhelpforum on 17th April 2011.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors