Jtdylktuy

Here’s one to get things started.

Let $U$ be a non-empty open subset of $Bbb R$. For $x,yin U$ define $xsim y$ iff $big[min{x,y},max{x,y}big]subseteq U$. It’s easily checked that $sim$ is an equivalence relation on $U$ whose equivalence classes are pairwise disjoint open intervals in $Bbb R$. (The term interval here includes unbounded intervals, i.e., rays.) Let $mathscr{I}$ be the set of $sim$-classes. Clearly $U=bigcup_{I in mathscr{I}} I$. For each $Iinmathscr{I}$ choose a rational $q_Iin I$; the map $mathscr{I}toBbb Q:Imapsto q_I$ is injective, so $mathscr{I}$ is countable.

A variant of the same basic idea is to let $mathscr{I}$ be the set of open intervals that are subsets of $U$. For $I,Jinmathscr{I}$ define $Isim J$ iff there are $I_0=I,I_1,dots,I_n=Jinmathscr{I}$ such that $I_kcap I_{k+1}nevarnothing$ for $k=0,dots,n-1$. Then $sim$ is an equivalence relation on $mathscr{I}$. For $Iinmathscr{I}$ let $[I]$ be the $sim$-class of $I$. Then $left{bigcup[I]:Iinmathscr{I}right}$ is a decomposition of $U$ into pairwise disjoint open intervals.

Both of these arguments generalize to any LOTS (= Linearly Ordered Topological Space), i.e., any linearly ordered set $langle X,lerangle$ with the topology generated by the subbase of open rays $(leftarrow,x)$ and $(x,to)$: if $U$ is a non-empty open subset of $X$, then $U$ is the union of a family of pairwise disjoint open intervals. In general the family need not be countable, of course.

These answers all seem to be variations on one another, but I've found each one so far to be at least a little cryptic. Here's my version/adaptation.

Let $U subseteq R$ be open and let $x in U$. Either $x$ is rational or irrational. If $x$ is rational, define
begin{align}I_x = bigcuplimits_{substack{Itext{ an open interval} \ x~in~I~subseteq~U}} I,end{align}
which, as a union of non-disjoint open intervals (each $I$ contains $x$), is an open interval subset to $U$. If $x$ is irrational, by openness of $U$ there is $varepsilon > 0$ such that $(x - varepsilon, x + varepsilon) subseteq U$, and there exists rational $y in (x - varepsilon, x + varepsilon) subseteq I_y$ (by the definition of $I_y$). Hence $x in I_y$. So any $x in U$ is in $I_q$ for some $q in U cap mathbb{Q}$, and so
begin{align}U subseteq bigcuplimits_{q~in~U cap~mathbb{Q}} I_q.end{align}
But $I_q subseteq U$ for each $q in U cap mathbb{Q}$; thus
begin{align}U = bigcuplimits_{q~in~U cap~mathbb{Q}} I_q, end{align}
which is a countable union of open intervals.

In a locally connected space $X$, all connected components of open sets are open. This is in fact equivalent to being locally connected.

Proof: (one direction) let $O$ be an open subset of a locally connected space $X$. Let $C$ be a component of $O$ (as a (sub)space in its own right). Let $x in C$. Then let $U_x$ be a connected neighbourhood of $x$ in $X$ such that $U_x subset O$, which can be done as $O$ is open and the connected neighbourhoods form a local base. Then $U_x,C subset O$ are both connected and intersect (in $x$) so their union $U_x cup C subset O$ is a connected subset of $O$ containing $x$, so by maximality of components $U_x cup C subset C$. But then $U_x$ witnesses that $x$ is an interior point of $C$, and this shows all points of $C$ are interior points, hence $C$ is open (in either $X$ or $O$, that's equivalent).

Now $mathbb{R}$ is locally connected (open intervals form a local base of connected sets) and so every open set if a disjoint union of its components, which are open connected subsets of $mathbb{R}$, hence are open intervals (potentially of infinite "length", i.e. segments). That there are countably many of them at most, follows from the already given "rational in every interval" argument.

Let $Usubseteqmathbb R$ open. Is enough to write $U$ as a disjoint union of open intervals.

For each $xin U$ we define $alpha_x=inf{alphainmathbb R:(alpha,x+epsilon)subseteq U, text{ for some }epsilon>0}$ and $beta_x=sup{betainmathbb R:(alpha_x,beta)subseteq U}$.

Then $displaystyle U=bigcup_{xin U}(alpha_x,beta_x)$ where ${(alpha_x,beta_x):xin U}$ is a disjoint family of open intervals.

A variant of the usual proof with the equivalence relation, which trades in the ease of constructing the intervals with the ease of proving countability (not that either is hard...):

1. Define the same equivalence relation, but only on $mathbb Q cap U$:
   $q_1 sim q_2$ iff $(q_1, q_2) subset U$ (or $(q_2, q_1) subset U$, whichever makes sense).


2. From each equivalency class $C$, produce the open interval $(inf C, sup C) subset U$ (where $inf C$ is defined to be $-infty$ in case $C$ is not bounded from below, and $sup C = infty$ in case $C$ is not bounded from above).


3. The amount of equivalence classes is clearly countable, since $mathbb Q cap U$ is countable.

Let $U$ be an open subset of $mathbb{R}$. Let $P$ be the poset consisting of collections $mathcal{A}$ of disjoint open intervals where we say $mathcal{A} le mathcal{A}'$ if each of the sets in $mathcal{A}$ is a subset of some open interval in $mathcal{A}'$. Every chain $C$ in this poset has an upper bound, namely $$mathcal{B} = left{ bigcupleft{J in bigcupbigcup C : I subseteq J right}: I in bigcupbigcup Cright}.$$
Therefore by Zorn's lemma the poset $P$ has a maximal element $mathcal{M}$. We claim that the union of the intervals in $mathcal{M}$ is all of $U$. Suppose toward a contradiction that there is a real $x in U$ that is not contained in any of the intervals in $mathcal{M}$. Because $U$ is open we can take an open interval $I$ with $x in I subseteq U$.
Then the set
$$mathcal{M}' = {J in mathcal{M} : J cap I = emptyset} cup left{I cup bigcup {J in mathcal{M} : J cap I ne emptyset}right}$$
is a collection of disjoint open intervals and
is above $mathcal{M}$ in the poset $P$, contradicting the maximality of $mathcal{M}$. It remains to observe that $mathcal{M}$ is countable, which follows from the fact that its elements contain distinct rational numbers.

Note that the only way in which anything about order (or connectedness) is used is to see that $I cup bigcup {J in mathcal{M} : J cap I ne emptyset}$ is an interval.

Let $G$ be a nonempty open set in $mathbb{R}$. Write $asim b$ if the closed interval $[a, b]$ or $[b, a]$ if $b<a$, lies in $G$.This is an equivalence relation, in particular $asim a$ since ${a}$ is itself a closed interval. $G$ is therefore the union of disjoint equivalence classes.

Let $C(a)$ be the equivalence class containing $a$. Then $C(a)$ is clearly an interval. Also $C(a)$ is open, for if $kin C(a)$, then $(k-epsilon, k+epsilon)subseteq G$ for sfficiently small $epsilon$.

But then $(k-epsilon, k+epsilon)subseteq C(a)$; so $G$ is the union of disjoint intervals. These are at most countable in number by Lindel$ddot{rm o}$f's theorem. This completes the proof.

Reference:G. De Barra, Measure theory and Integration, Horwood Publishing.

(http://infoedu.ir/wp-content/uploads/2014/03/MeasureTheoryBook.pdf)

$mathbb{R}$ with standard topology is second-countable space.

For a second-countable space with a (not necessarily countable) base, any open set can be written as a countable union of basic open set.

Given any base for a second countable space, is every open set the countable union of basic open sets?

Clearly, collection of open intervals is a base for the standard topology.
Hence any open set in $mathbb{R}$ can be written as countable union of open intervals.

If any two of exploited open intervals overlap, merge them. Then we have disjoint union of open intervals, which is still countable.

This proof is an extended version of the nice proof proposed by
Stromael and it serves best for beginners who want to understand every detail(that one that for any established mathematician logically seems trivial) of the proof.

$ textbf{Proof:} $

Let $U subseteq R$ be open and let $x in U$. Then Either $x$ is rational or $x$ is irrational.

Suppose $x$ is rational, then define

begin{align} I_x = bigcuplimits_{substack{Itext{ an open interval} \ x~in~I~subseteq~U}} I,end{align}

Claim: $I_x$ is interval, $I_x$ is open and $ I_x subseteq U $

Definition: An interval is a subset $ I subseteq R$ such that, for all $ a<c<b$ in $R$, if $ a,b in I $ then $ c in I$.

Now, consider any $ a<c<b $ such that $ a,b in I_x$. We want to show that $ c in I_x $.

Denote $I_a $ to be an interval such that $ x in I_a $ and $ a in I_a $. In other words $ I_a $ is one of the intervals from the union $ I_x $ that contains $a$. In the same way, let $ I_b $ be the interval such that $ x in I_b $ and $ b in I_b $.

1. $ c=x $: If $c=x$ then by construction of $I_x$, $ x in I_x$




2. $ c<x $: If $c<x$ then we have that either $ a<c<x<b $ or $ a<c<b<x $. Since $ x in I $ for every open interval $I$ of the union $I_x$ (by construction of $I_x$ ), we have that $x in I_a $ and $ x in I_b$. Since $ x in I_a $ then because $ I_a $ is an interval $ c in I_a$ and hence $ c in I_x $. And since $ x in I_b $ then because $ I_b $ is an interval $ c in I_b $ and hence $ c in I_x $. Thus, we concluded that $ c in I_x $.




3. $ c > x $: If $ c>x $ then we have that either $ a<x<c<b $ or $ x<a<c<b $. Since $ x in I $ for every open interval $I$ of the union $I_x$ (by construction of $I_x$ ), we have that $x in I_a $ and $ x in I_b$. Since $ x in I_b $ then because $ I_b $ is an interval $ c in I_b $ and hence $ c in I_x $. As for the second case, note that since $ x in I_b$ we have that $ a in I_b $. But then, because $ I_b $ is an interval we have that $ c in I_b $ and hence $ c in I_x$. Hence we concluded that $ c in I_x $.

This Proves that $ I_x $ is an interval.

$ I_x $ is open because it is union of open sets.

$ I_x subseteq U $ by construction.

Suppose $x$ is irrational, then by openness of $ U $ there is $varepsilon > 0$ such that $(x - varepsilon, x + varepsilon) subseteq U$, and by the property of real numbers that for any irrational number there exists a sequence of rational unmbers that converges to that irrational number, there exists rational $y in (x - varepsilon, x + varepsilon) $. Then by construction $ (x - varepsilon, x + varepsilon) subseteq I_y $. Hence $x in I_y$. So any $x in U$ is in $I_q$ for some $q in U cap mathbb{Q}$, and so

begin{align}U subseteq bigcuplimits_{q~in~U cap~mathbb{Q}} I_q.end{align}

But $I_q subseteq U$ for each $q in U cap mathbb{Q}$; thus

begin{align}U = bigcuplimits_{q~in~U cap~mathbb{Q}} I_q, end{align}

which is a countable union of open intervals.

Now let's show that intervals $ {I_q } ~ q in U cap mathbb{Q} $ are disjoint. Suppose there is $ i, j, in U cap mathbb{Q} $ such that $ I_i cap I_j neq emptyset $ then $ I_i subseteq I_q $ and $ I_j subseteq I_q $ for some $ q in U cap mathbb{Q} $

Hence we constructed disjoint intervals $ {I_q } ~ q in U cap mathbb{Q} $ that are enumerated by rational numbers in $U$ and whose union is $U$. Since any subset of rational numbers is countable, $ {I_q } ~ q in U cap mathbb{Q} $ is countable as well. This finishes the proof.

I hope that this is right as this is a lemma i've thought of and i plan to use in a project due in several days and it somewhat generalizes the question asked:

Suppose that $U$ is a set of intervals in $mathbb{R}$ (closed, open, semi-closed, etc.). Then there exists a set of disjoint intervals $V$ in $mathbb{R}$ s.t. $bigcup_{Iin U}I=biguplus_{Iin V}I$. If non of the intervals are degenerate or $U$ is countable, then this set can be taken to be countable. And if they were all open, then we could take the segments of $V$ to be open (And also, if $U$ is countable then we won't be needing the Axiom of Choice).

proof:
Let us order the elements of $U$: $U=langle I_beta,|,betaleqalpharangle$ where $alpha$ is the first ordinal of cardinality $|U|$. (If $U$ is countable then this doesn't require AC and from here on will be standard induction with a simple construction in the end for $omega$).

I'll build $V_beta=langle J^gamma_beta,|,gammaleqbetarangle$ - a sequence of segments for all $betaleqalpha$ such that every two sets in $V_beta$ are either disjoint or equal and such that $displaystyle{bigcup_{gammaleqbeta}I_gamma=biguplus_{gammaleqbeta}J^gamma_beta}$ and $forallbeta$, $langle J^beta_gammarangle_{gammageqbeta}$ is a non-descending sequence of sets ,by means of transfinite induction.

For $V_0$ take, $V_0=langle I_0rangle$. Suppose that we have built the required $V_gamma$, $gamma<beta$ for some $betaleqalpha$, then we will build $V_beta$ in the following way: $forallgamma<beta$, denote $widetilde{J}_gamma$=$bigcup_{gammaleqdelta<beta}J_delta^gamma$-still segments (non-decreasing sequence). If $I_beta$ is disjoint of all $widetilde{J}_gamma$, taking $V_beta!=!langle widetilde{J}_gamma,|,gamma<betaranglecup{(beta,I_beta)}$ would give us a sequence $langle V_gamma,|,gammaleqbetarangle$ satisfying the required conditions of it (the only non trivial thing is that pairs of $widetilde{J}_gamma$ are either disjoint of each other or are equal, but that is also quite trivial since if the contrary would have occurred, then $existsgamma_1<gamma_2<beta$ s.t. $widetilde{J}_{gamma_1}neqwidetilde{J}_{gamma_2}$ and $widetilde{J}_{gamma_1}capwidetilde{J}_{gamma_2}neqemptyset$, but then, $exists beta>delta_1geqgamma_1, beta>delta_2geqgamma_2$ s.t. $J^{gamma_1}_{delta_1}cap J^{gamma_2}_{delta_2}neqemptyset$, meaning that either $J^{gamma_1}_{delta_2}= J^{gamma_2}_{delta_2}$ or $J^{gamma_1}_{delta_1}= J^{gamma_2}_{delta_1}$ thus, $forallbeta>epsilongeqdelta_1,delta_2$, $J^{gamma_1}_{epsilon}= J^{gamma_2}_{epsilon}$ and since we are talking here about non-decreasing sequences, this will contradict $widetilde{J}_{gamma_1}neqwidetilde{J}_{gamma_2}$). And if $I_beta$ isn't disjoint of all $widetilde{J}_gamma$, Then we can take $J_beta^gamma=widetilde{J}_gamma$ for all $gamma<beta$ that don't intersect with $I_beta$ and $J_beta^gamma=bigcup_{delta<betatext{ s.t. }widetilde{J}_deltacap I_betaneqemptyset}{widetilde{J}_delta}cup I_beta$ - segment for all of the other $gammaleqbeta$. Then again from the same arguments, $langle V_gamma,|,gammaleqbetarangle$ would satisfy the required conditions.

Finally, we can take $V={J_alpha^beta,|,betaleqalpha}$ to get what we wanted in the first place. And obviously, if our segments were all non-degenerate to begin with, from the way we constructed our set, all of the segments in $V$ will be non-degenerate (and thus of positive measure), but they are disjoint and so there is only a countable number of them. And if the segments in $U$ were all open, then obviously, so will the segments in $V$.$square$

The balls with radii $frac{1}{n}$ and center at a rational number form a basis for the euclidean topology. This family is countable as it is a countable union of countable sets. We have found a countable basis, so we are done.

Essentially nothing differs here from the two previous responses which rely principally on the fact that $mathbb{R} $ is locally connected. However I present here a proof that hopefully will feel accessible to readers with a slightly lower level of topological literary at the cost of appearing cumbersome to an expert.

Consider the connected components of $U$.
$U_x subseteq U$ is defined to be the connected component of U containing $x$ if $U_x$ is the largest connected subset of $U$ which contains $x$.
Clearly by definition $U_x=U_v$ if $v in U_x$.
Therefore if $U_a cap U_b neq varnothing$ then $U_a=U_b$.
We see that ${U_x}_{xin U}$ is a disjoint collection.
Also it should be clear that $bigcup limits_{xin U} U_x = U$.

Now we show that $forall x$ $U_x$ is open.
Let $yin U_x subseteq U$.
Since $U$ is open there exists $epsilon>0$ such that $(y-epsilon, y+epsilon )subseteq U$.
Sets of real numbers are connected iff they are intervals, singletons or empty.
$(y-epsilon,y+epsilon)$ an interval hence it is connected.
Therefore since $U_y$ is the largest connected subset of $U$ containing $y$ we must have $(y-epsilon,y+epsilon)subseteq U_y =U_x$.
This shows that $U_x$ is open for all $x$.

$U_x$ open and connected implies that $U_x$ must be an open interval.

Also $mathbb{Q}$ dense in $mathbb{R}$,
so $forall xin U$, $U_xcap mathbb{Q}neq varnothing$ and $U_x=U_q$ for some $qinmathbb{Q}$.
So we can write ${U_x}_{xin U}={U_q}_{qin S}$ for some $Ssubseteq mathbb{Q}$.
$mathbb{Q}$ is countable so $S$ is at most countable.

In conclusion, we have just shown that the union of the connected components of $U$ is a disjoint union of open intervals that equals $U$ and is at most countable.

The proof that every open set is a disjoint union of countably many open intervals relies on three facts:

• $Bbb R$ is locally-connected


• $Bbb R$ is ccc


• The open connected sets in $Bbb R$ are open intervals

Let $Usubseteq Bbb R$ be open. Then there is a collection of disjoint, open, connected sets ${G_alpha}_{alphain A}$ such that $U=bigcup_{alphain A} G_alpha$. Since $Bbb R$ is ccc, the collection ${G_alpha}$ is at most countable. Since the open connected sets $Bbb R$ are open intervals, ${G_alpha}$ is a countable collection of disjoint, open intervals.

The first two facts allow us to see some generalizations. Namely any open set in a locally-connected, ccc space is a countable disjoint union of connected open sets. This applies to any Euclidean space. Although open connected subsets of Euclidean space are more complicated than open intervals, they are still relatively well-behaved.

The following is certainly not the quickest approach to a proof, but when this question was first posed to me in class, my first intuition was to use some elementary graph theory:

Let $U$ be an open set of $mathbb{R}$. As we know, $mathbb{R}$ has a countable basis $mathcal{B}$ comprised of connected open sets and so we may write $U=bigcup_{nin I} U_n$, where for each $n$ we have $U_ninmathcal{B}$ and $I$ is some countable index set.

Let $G$ be the intersection graph of ${U_n}$. That is to say, the vertex set of $G$ is simply ${U_n}$ and there is an edge between $U_i$ and $U_j$ iff they have nonempty intersection. It's easy to convince yourself that:

• This graph must have countably many graphically-connected components (otherwise we'd have uncountably many vertices which is impossible).


• The intersection graph of $Asubseteq{U_n}$ is graphically-connected iff for any two $V,Win A$ there is a sequence $V=U_{n_1},U_{n_2},ldots,U_{n_k}=W$ such that $U_{n_i}cap U_{n_{i+1}}neqvarnothing$.


• The union $bigcup A$ is a connected set of $mathbb{R}$ whenever the intersection graph of $A$ is graphically-connected.

Thus, when we take the union of all the vertices within a graphically-connected component, for every component, we obtain countably-many connected open sets. The union of these sets is of course $U$ itself. Since the connected open sets of $mathbb{R}$ are intervals (including rays), we're done.

Side Note: This would also work in $mathbb{R}^n$ or in general, any topological space $X$ that has a countable basis comprised of connected sets. Well, so long as we replace countable union of disjoint open intervals with countable union of disjoint open connected sets.

The balls with radii $frac{1}{n}$ and center at a rational number form a basis for the euclidean topology. This family is countable since $mathbb N times mathbb Q equiv mathbb N$ and we have a countable basis $(B_lambda)_{ , lambda in mathbb N times mathbb Q}$ of open intervals for $mathbb R$.

Let $U$ be a nonempty open set in $mathbb R$; we can express it as countable union of open balls from $(B_lambda)$.
Also,
$tag 1 text{ }$
$quad$ If two open intervals have a nonempty intersection, then their union is also an open interval.

So if $U$ is a finite union of the $B_lambda$, it is an easy matter to combine the $B_lambda$, if necessary, and writing $U$ as a disjoint union of a finite number of open intervals. So assume, WLOG, that

$tag 2 U = bigcup_{, n in mathbb N ,} B_n$.

We define a relation on our (new) index set $mathbb N$ with $msim n$ if $B_m cap B_n ne emptyset$ or there is is a finite 'nonempty intersection $Btext{-}$chain' connecting $B_m$ with $B_n$. It is easy to see that this partitions $mathbb N$ and that taking the corresponding unions of the $B_n$ over an index $lambda text{-}$ block gives a partition of $U$. Also, using (1), we can show that we have also expressed $U$ as a countable union of disjoint open intervals.

More of a question than answer. I am a chemist turned pharmacist, who wishes to have studied mathematics. I am trying to work through Rudin's Principles of Mathematical Analysis. I envy you all who are involved in math for a career.

Can someone give me feedback on my attempt at a proof in $mathbb R? Completely novice and not at all pretty, but is it sound?

segment := open interval in R.

Lemma: disjoint segments in R are separated (proof not shown).

It follows from the lemma that an open connected subset of R cannot be the union of disjoint segments.

Let E be an open subset of R. Since R is separable by Rudin prob 2.22, there exists a subset, D, of R that is countable and dense in R. Assume E is connected, which includes E=R, then E is the union of an at most countable collection of open segments, containing only E.

Suppose E is separated. Then E is the union of a collection of disjoint segments, including the possibility of segments unbounded above or below.

If the collection is finite, then it is at most countable.

Assume the collection of segments is infinite. Because D is dense in R and E is contained in R, every open subset of E contains a point of D. Then each of the infinitely many disjoint segments contains a unique point of D. Since D is countable, a one-to-one correspondence between a unique point of D and the segment is established. This implies that there are countably many disjoint segments in the collection.

Therefore E is the union of finitely many or countably many
, hence at most countably many, disjoint segments.