Closed and Open Subsets

**selinunan** · December 20th 2008, 04:40 PM

If anyone helps me with these questions, I will be thankful.

(a) Show that every closed set in a metric space can be expressed as an
intersection of countably many open sets.

(b) Show that every open set in a metric space can be expressed as an
union of countably many closed sets. (Hint: take complements.)

(c) Express the interval (0, 1) in R as a union of countably many open
sets.

**kalagota** · December 20th 2008, 05:25 PM

Originally Posted by selinunan

(c) Express the interval (0, 1) in R as a union of countably many open
sets.

$(0,1) = \bigcup_{n\in\mathbb{N}} \left(\dfrac{1}{n+1},1\right)$

**selinunan** · December 20th 2008, 06:15 PM

Thank you for your help.

Will it be OK if I answer the part (a) like this?

Let A_t be an open set in (M,d).

Then, A_t = { d (a,x) < t for some a in A }

So, ∩ { d (a,x) < 1/n for some a in A } = cl (A) which is the smallest closed set containing A.

If it is true, how can I modify it for the answer of part (b)?

**kalagota** · December 20th 2008, 08:43 PM

i think, you should start with an arbitrary closed set and find a way that it can be expressed as an intersection of countably many open sets..

what you did is you take an arbitrary open set and come up with a closed set related to that..

**selinunan** · December 20th 2008, 09:14 PM

Do you have an idea how I can manage to do it?

Thank you.

**kalagota** · December 20th 2008, 09:34 PM

honestly, this is the first time i encountered such question.. so i'm still trying to device a proof for it..

**Plato** · December 21st 2008, 04:10 AM

Here is a start. Suppose that $M$ is a closed set.
$\left( {\forall n \in \mathbb{Z}^ + } \right)$ define $<br /> O_n = \bigcup\limits_{x \in M} {B\left( {x;\frac{1}{n}} \right)}$ .

**selinunan** · December 21st 2008, 07:29 AM

Thank you, but I couldn't understand how I can go on.

**Plato** · December 21st 2008, 07:51 AM

Originally Posted by selinunan

I couldn't understand how I can go on.

Well each $O_n$ is an open set being a union of balls and $\left( {\forall n} \right)\left[ {M \subseteq O_n } \right]$ .
Thus $M \subseteq \bigcap\limits_{n = 1}^\infty {O_n }$ . To show equality suppose that $y \notin M$ .
Because of closure, $\left( {\exists \delta > 0} \right)\left[ {B(y;\delta ) \cap M = \emptyset } \right]$ .
$\left( {\exists N} \right)\left[ {\frac{1}{N} < \delta } \right]$ so is it possible for $y \in B\left( {x;\frac{1}{N}} \right) \subseteq O_N$ .
If not then $M = \bigcap\limits_{n = 1}^\infty {O_n }$ .

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