Topology – Union of Closure of Sets: Finite vs Infinite Unions

general-topologymetric-spaces

Let $A_i$ be a subset of a metric space for each $i\in \mathbb{N}$.

  1. Let $B_n := \bigcup_{i=1}^n A_i$. Prove (for any) $n \in \mathbb{N}$ that $\overline{B_n} = \bigcup_{i=1}^n \overline{A_i}$.
  2. If $B = \bigcup_{i=1}^\infty A_i$, prove that $\overline{B} \supseteq \bigcup_{i=1}^\infty \overline{A_i}$. Give an example to show that this containment might be proper.

If $A_i$ is closed then $A_i = \bar A_i$, but I'm stuck as to how to prove $B=\bar B_n$. If I prove the first statement for when $A_i$ is closed does that mean it is also true for when $A_i$ is open because I can construct a closed set containing $A_i$?

For the example, would constructing a sequence of closed segments between $0$ and $1$ that gets arbitrarily close to $1$ and taking the union of the segments be considered a proper containment?

Best Answer

To prove the true statements we'll make use of the following facts:

  • If $A \subseteq B$, then $\overline{A} \subseteq \overline{B}$.
  • $x \in \overline{A}$ iff $U \cap A \neq \varnothing$ for every open neighbourhood $U$ of $x$. (This is in the more general context of general topological spaces. For metric spaces it suffices to restrict the $U$s to the open $\varepsilon$-balls centred at $x$.)

To see that $\bigcup_{i=1}^n \overline{A_i} \subseteq \overline{B_n}$ observe that $A_i \subseteq B_n$ for each $i \leq n$, and apply the first fact above.

To see that $\overline{B_n} \subseteq \bigcup_{i=1}^n \overline{A}_i$ note that this is equivalent to showing $X \setminus \bigcup_{i=1}^n \overline{A}_i \subseteq X \setminus \overline{B_n}$. If $x \in X \setminus \bigcup_{i=1}^n \overline{A_i}$, then by the second fact above it follows that for each $i \leq n$ there is an open neighbourhood $U_i$ of $x$ which is disjoint from $A_i$. Now note that $U := U_1 \cap \cdots \cap U_n$ is an open neighbourhood of $x$ which is disjoint from $B_n$.

Note that the first fact above also shows that $\bigcup_{i=1}^\infty \overline{A_i} \subseteq \overline{\bigcup_{i=1}^\infty A_i} = \overline{B}$.

To give an example of where the equality does not hold, just enumerate the set $\mathbb{Q}$ of rational numbers as $\{ a_i : i \in \mathbb{N} \}$, and for each $i$ define $A_i = \{ a_i \}$. Working in $\mathbb{R}$ with the usual (metric/order) topology, we can show that $$ {\textstyle \bigcup_{i=1}^\infty} \overline{A_i} = \mathbb{Q} \neq \mathbb{R} = \overline { {\textstyle \bigcup_{i=1}^\infty} A_i }. $$

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