General Topology Problem Solution Engelking |verified| Jun 2026

Let ( X ) be a topological space, ( A \subset X ). Prove that ( \partial (\partial A) \subset \partial A ). Find an example where ( \partial (\partial A) \neq \partial A ).

Engelking assumes a high level of mathematical maturity. Problems frequently rely on set theory (Zorn's Lemma, ordinal numbers, cardinal arithmetic) as if it were second nature to the reader. If your set theory is rusty, the topology problems will feel impossible, not because of the topology, but because of the foundations. General Topology Problem Solution Engelking

Let us dissect a problem that appears in every "Help me with Engelking" forum post: (in the 1989 edition). Prove that for a normal space $X$, the following are equivalent: (a) $X$ is perfectly normal; (b) Every open set is an $F_\sigma$. Let ( X ) be a topological space, ( A \subset X )

This is simple, yet half the online "solutions" forget to state that perfect normality includes normality as a hypothesis, leading to a circular argument. Engelking assumes a high level of mathematical maturity