Countable Intersection of Open Sets Continuity Preimage
Closed set as a countable intersection of open sets
Solution 1
Let $A\subseteq X$ be closed. For all $n\in \mathbb N$ define $$U_n=\bigcup _{a\in A} B(a,\frac{1}{n}).$$ $U_n$ is open as a union of open balls. We prove that $A=\bigcap _{n\in \mathbb N} U_n$.
Clearly $A \subseteq \bigcap _{n\in \mathbb N} U_n$.
To prove $A \supseteq \bigcap _{n\in \mathbb N} U_n$ we take $x\notin A$ and show that $x\notin \bigcap U_n$.
Since $A$ is closed, $A^C$ is open, therefore $\exists n \in \mathbb N$ such that $B(x,\frac{1}{n}) \cap A=\emptyset$. That is, for all $a\in A$: $a\notin B(x,\frac{1}{n})$; and thus for all $a\in A$: $x\notin B(a,\frac{1}{n}) \Longrightarrow x\notin \bigcup_{a\in A} B(a,\frac{1}{n}) \Longrightarrow x\notin U_n \Longrightarrow x\notin \bigcap U_n$.
Solution 2
Hint: Let $A\subseteq X$ be closed. For $\epsilon > 0$ let $$ U_\epsilon(A) = \{x \in X \mid \def\dist{\mathop{\rm dist}}\dist(x,A) < \epsilon \}$$ where $\dist(x, A) := \inf_{y \in A} d(x,y)$. What can you say about the sets $U_\epsilon(A)$, what is $\bigcap_{\epsilon > 0} U_\epsilon(A)$?
Solution 3
Let $D(A,\varepsilon)=\left\{ y\in M | d(A,y)<\varepsilon \right\}$, where $d(A,y)=\inf \left\{ d(z,y)|z\in A \right\}$. This set is open .Define a sequence of $\varepsilon$ as $\varepsilon_{n}=\frac{1}{n}$. Claim $A=\cap_{n=1}^{\infty}D(A,\varepsilon_{n})$. Proof: If $x\in A$, then $d(A,x)=0$ and so $x \in D(A,\varepsilon_{n})$ for all $n\in \mathbb{N}$ $\Rightarrow$ $A\subset \cap_{n=1}^{\infty}D(A,\varepsilon_{n})$. Conversely if $x\in \cap_{n=1}^{\infty}D(A,\varepsilon_{n})$ then $x\in D(A,\varepsilon_{n})$ for all $n\in \mathbb{N}$ $\Rightarrow$ $\forall \varepsilon>0$ $D(x,\varepsilon)\cap A\setminus \left\{ x \right\}\neq \emptyset$ and so $x$ is an accumulation point of $A$. $A$ is, however, closed so contains all its accumulation points, so $x\in A$ $\Rightarrow$ $\cap_{n=1}^{\infty}D(A,\varepsilon_{n})\subset A$.
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Comments
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Let's take a metric space. Then any closed set can be written as a countable intersection of open sets.
How can I prove that?
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Linking to another related question with answer: math.stackexchange.com/questions/1444562/… (In a metric space, each closed set is a countable intersection of open sets and each open set is a countable union of closed sets.)
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I think this approach makes it a little difficult to see that the $U_\epsilon$ are open. Defining $U_\epsilon$ to be the union of open balls of radius $\epsilon$ around points of $A$ would make its openness immediate.
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As $\mathop{\rm dist}(\cdot, A) \colon X \to [0,\infty)$ is continuous ...
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True, but more elementary is that a union of open sets is open.
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You are right, but on the other hand, the way I used above is how I imagine the set $U_\epsilon(A)$ in the first place, namely as $A$ with a small border around it, not as a union of balls.
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l-2: Isn't the $D(x,\epsilon)$ meant to be a ball centered at x ?
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it is not clear to me why $A \subseteq \bigcap_{n \in \mathbb{N}}U_{n}$, could you please explain this part in more detail?
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@JessyCat: well, let's take an element $a\in A$ and show that it's in $\bigcap U_n$. Note that for every $n\in \mathbb N$ we have $a\in B(a,\frac{1}{n})$, and thus $a\in U_n$ for every $n\in \mathbb N$. This means that $a\in \bigcap U_n$. Hope it helped :)
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let's say we were talking about $\mathbb{R}$ here, instead of a generalized metric space. What would an element of $A$ look like then? The problem would reduce to showing that a closed interval in $\mathbb{R}$ equals the countable intersection of open intervals in $\mathbb{R}$, so I would like to see in more detail what an element of a closed interval in $\mathbb{R}$ looks like. Is it just a point?
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Not every closed set in $\mathbb R$ is an interval, so we can't reduce the problem this way. But we can prove, in particular, that every closed interval is a countable intersection of open intervals. For this purpose, the elements of the intervals in question are indeed points.
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also, how does $A^{c} \notin \bigcap U_{n}$ necessarily imply that $\bigcap U_{n} \subseteq A$?
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it doesn't, and I didn't say it does...
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@Ludolila can I ask you to explain me one thing: why do we need to define the union of open balls?
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@DanielYefimov I'm not sure I understand your question. We want $A$ to be a countable intersection of open sets. We define these open sets, $U_n$, as unions of (open) neighborhoods of elements of $A$. In a metric space, we can work with open balls, instead of some general neighborhoods...
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@Ludolila Thank you, now I got it:) Just forgot, that every open set is a union of neighborhoods of points of open set.
Recents
Source: https://9to5science.com/closed-set-as-a-countable-intersection-of-open-sets
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