I'm minutely studying "Counterexamples in Topology" by Steen and Seebach (great book! Recommended) and have reached a statement on Page 7 which I can't prove.

"The

**exterior** ext(A) of a set A is the complement of the closure of A, or equivalently, the interior of the complement of A. ..."

(The closure is the set together with its limit points, the interior is conveniently the complement of the closure of the complement.)

"... the exterior of the intersection is contained in the union of the exteriors ... equality holds for finite unions ..."

So my approach is as follows:

Let T be a topological space and let \mathbb H \subseteq P \left({T}\right) where P \left({T}\right) is the powerset of T.

We want to show ext\left({\bigcap_{H \in \mathbb H} H}\right) \subseteq \bigcup_{H \in \mathbb H} ext(H).

We use cl(H) as the closure of H.

We have by definition of exterior:

ext \left({\bigcap_{H \in \mathbb H} H}\right) = T - cl \left({\bigcap_{H \in \mathbb H} H}\right)

But we have that:

cl \left({\bigcap_{H \in \mathbb H} H}\right) \subseteq \bigcap_{H \in \mathbb H} cl \left({H}\right)

from

Closure of Intersection Subset of Intersection of Closures - ProofWiki
But then A \subseteq B \Longrightarrow T - B \subseteq T - A and so:

T - cl \left({\bigcap_{H \in \mathbb H} H}\right) \supseteq T - \bigcap_{H \in \mathbb H} cl \left({H}\right)

By de Morgan this gives:

T - \bigcap_{H \in \mathbb H} cl \left({H}\right) = \bigcup_{H \in \mathbb H} cl \left({T - H}\right)

which by definition of exterior equals \bigcup_{H \in \mathbb H} ext (H)

Putting it all together it seems I've just proved:

ext \left({\bigcap_{H \in \mathbb H} H}\right) \supseteq \bigcup_{H \in \mathbb H} ext (H)

which is the antithesis of what I was expecting to get.

Where am I going wrong - or are Steen and Seebach wrong?

BLAST IT - The LaTeX won't work. Help might be needed.