2
votes
1answer
33 views

If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?

I was working my way through the set theory chapter in my Analysis textbook when I stumbled across these two theorems: Every infinite set has a countable subset A union of countable subsets ...
3
votes
4answers
104 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). ...
0
votes
3answers
48 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
2
votes
1answer
54 views

Generalizing the Monotone Subsequence theorem

In proving the Bolzaono-Weierstrass theorem, one proves the lemma that every infinite real sequence has a(n infinite) monotone subsequence. In all of the proofs I've seen so far, this is done by ...
1
vote
0answers
35 views

On $\sigma$-algebra generated by sets

Given $\mathcal{S}$ a collection of subsets of $X$ and $A\subset X$. To show that $\sigma(\mathcal{S}\cap A)=\sigma(S)\cap A$, where for any collection of $\mathcal C$ of subsets of $X$, $\mathcal ...
1
vote
3answers
72 views

Proving the open interval $(0,1)$ is uncountable [duplicate]

I am currently able to prove this statement using the Cantor diagonalisation argument, my question is whether there is another way (more simple or more complex) to prove this statement, without ...
1
vote
1answer
30 views

How to embed a total ordering into the real field.

Let $(S,<_S)$ be a total ordering with $card(S)\leq card(2^{\aleph_0})$. Does there exist a subset $A$ of the real numbers such that $(A,<_A)$, being a total ordering, is isomorphic to ...
11
votes
4answers
183 views

How find this minimum of the value $f(1)+f(2)+\cdots+f(100)$

Give the positive integer set $A=\{1,2,3,\cdots,100\}$, and define function $f:A\to A$ and (1):such for any $1\le i\le 99$,have $$|f(i)-f(i+1)|\le 1$$ (2): for any $1\le i\le 100$,have ...
-1
votes
3answers
71 views

How many real valued Cauchy sequences are there? [closed]

Is the set of all Cauchy sequences of real numbers countable or uncountable? In other words, is $S$ countable or uncountable, where $$S=\big\{\langle x_{n}\vert ...
1
vote
3answers
47 views

Enumeration of rational numbers

If $\Bbb Q=\{q_n:n\in \Bbb N\}$ be an enumeration of $\Bbb Q$, is it true that $|q_n|<1/n$ for infinitely many $n$? I just come up with this question, it seemed simple but I can't solve it. Is ...
1
vote
2answers
26 views

Finite vs infinite distinction in Rudin's Analysis

I'm starting to self-study Rudin's Principles of Mathematical Analysis. I'm up to the second chapter, theorem 2.24. For any collection $\{G_i\}$ of open sets, $\bigcup_i^nG_i$ is open. For ...
0
votes
1answer
30 views

Sets A and C with m-1 Elements

If A is a set with m elements and C is a set with one element, then A-C is a set with m-1 elements. What is a proof for this statement?
0
votes
2answers
20 views

Proof Regarding Infinite Sets

If A is an infinite set and B is a finite set prove that A-B is an infinite set.
0
votes
1answer
23 views

Would this function be correct?

I want to express a function that for any x input, it outputs the nearest EVEN integer less than or equal to x. Would $g(x) = \{ \lfloor x \rfloor : 2 \mid x \} $ do the job properly? Read: $g(x)$ ...
0
votes
4answers
103 views

Suppose $f: X \to Y$ and $g: Y \to Z$ are onto functions. Show that $g \circ f: X \to Z$ is onto

I feel like my solution isn't sufficient at the moment. --> For some x in X there exists a y in Y given by $ f(x)=y $ because $f: X \to Y$ is onto. The same can be said for some y in Y existing as ...
5
votes
1answer
49 views

Show that $\mathfrak{Z}$ is a semi ring

Consider measurable spaces $(\Omega_t,\mathcal{A}_t), t\in T$ ($T$ is any index set). With $\mathcal{E}(T)$ we the set of all finite, not-empty subsets of $T$. Show that $$ ...
0
votes
0answers
40 views

Is there a diagram to help with these proofs? (sets and functions)

My book wrongly asserted that for any family $X_a$ of sets the following is true: $$f(\cap_aX_a)=\cap_af(X_a)$$ Because I could think of a counter example (y=$x^2$ - simple but works) I decided to ...
1
vote
1answer
59 views

Interior points and open sets

From what I understand, a set is open if every element or point in said set is an interior point. Now, suppose that I have a set $S$ with infinite points $s_0,s_1,s_k,...$ and so on. Mathematically, ...
0
votes
1answer
34 views

Transform this expression in something useful for me

I have $A \subset \Omega$ and $E \subset \Omega$. Now I have $A\cap E^C$. But I do not want to work with the complement, I am rather looking for an expression that somehow contains $A \cap E$. Does ...
2
votes
1answer
54 views

How to rigorously prove that these two sets have different order types?

Let $A$ and $B$ be two given ordered sets with the linear (or total) order relations $<_A$ and $<_B$, respectively. Then $(A,<_A)$ and $(B,<_B)$ are said to be of the same type if there ...
0
votes
0answers
66 views

Formal and general definition of natural domain (natural set) of a function

Can anyone give me a (as much as possible) formal and general definition of natural domain of a function? Let's say that a function is a triplet $(X,Y,f)$ where $f \subseteq X \times Y$ such that ...
1
vote
2answers
71 views

The complement of the closed subset of a closed set

Suppose that I had a closed set. Suppose that I made it so that there was nothing else outside such closed set. I will call this set "A". Now, suppose that I picked a closed subset from A. I will call ...
0
votes
2answers
47 views

$A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$.

Construct a sequence of measurable sets $A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$. My thoughts: Since $\lim\sup ...
0
votes
2answers
102 views

sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$.

Give an example in $\mathbb{R}$ of a sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$. My thoughts: By definition, ...
0
votes
1answer
44 views

Proving $m\lambda(E_m)\le\sum_{k=1}^{\infty}\lambda(A_k)$

Assume $A_1,A_2,...$ are measurable sets. Let $m\in\mathbb{N}$, and let $E_m$ be the set defined as follows: $x\in E_m\iff$ $x$ is a member of at least $m$ of the sets $A_k$. Prove that $E_m$ is ...
2
votes
1answer
70 views

cardinality of the set of all dense subsets of $\Bbb R$

Let $$A=\{X \subseteq \mathbb R : \operatorname{cl}(X)=\mathbb R\}$$ Prove that the set $A$ and $P(\mathbb R)$ have the same cardinality. Well, the first thing it came to my mind was the injective ...
1
vote
1answer
50 views

Suppose $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous functions. If $f$ and $g$ are measurable, then

Suppose $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous functions. If $f$ and $g$ are measurable, then $h(x) = F( f(x), g(x) ) $ is also measurable. Proof: For all $a$, $ \{ x : ...
2
votes
3answers
467 views

Set of irrationals between two reals is uncountable

I know that between any two reals, there is an irrational number. See: Proving that there exists an irrational number in between any given real numbers Now let a, b $\in$ $R$ such that a < b. And ...
0
votes
0answers
85 views

For $A\subset X$, prove that the characteristic function $\chi_A$ (which maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$.

For $A\subset X$, prove that the characteristic function $\chi_A$ (which maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$. I got the forward direction (i.e, proving $A\subset M$). For the ...
3
votes
1answer
75 views

A query about countability

Suppose we are working in an elementary context where we want to keep background assumptions modest (not take a stand on fancy issues in set theory, say). What should our attitude be to the idea of ...
1
vote
1answer
77 views

Describe explicitly the $M$-measurable functions in case $M$ is one of the following $\sigma$-algebras:

Describe explicitly the $M$-measurable functions in case $M$ is one of the following $\sigma$-algebras: (a) $M=\{\emptyset,X\}$ (b) $M=2^{X}$ (c) For certain disjoint sets $E_1,...,E_N$, ...
0
votes
0answers
64 views

Prove that a set $A\subset\mathbb{R}^n$ is measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$.

Prove that a set $A\subset\mathbb{R}^n$ is measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without ...
2
votes
1answer
155 views

Prove that if $N$ is a null set in $\mathbb{R}^n$, then there exists a Borel null set $N'$ such that $N\subset N'$.

Prove that if $N$ is a null set in $\mathbb{R}^n$, then there exists a Borel null set $N'$ such that $N\subset N'$. In fact, prove that $N'$ may be chosen to be a $G_{\delta}$, a countable ...
3
votes
2answers
811 views

Give an example of two $\sigma$ algebras whose union is not an algebra

Give an example of two $\sigma$ algebras in a set $X$ whose union is not an algebra. I've considered the sets $\{A|\text{A is countable or $A^c$ is countable}\}\subset2^\mathbb{R}$, which is a ...
0
votes
1answer
210 views

Expressing intervals as a union or intersection of intervals of the form $(a,b]$

I want to express all intervals as countable union or intersection of intervals of the form $(a,b]$. I already know $$ (a,b) = \bigcup_{n} (a, b - \frac{1}{n} ]$$ $$ [a,b] = \bigcup_{n} (a + ...
0
votes
1answer
106 views

A basis question of least upper bound property.

A set $A$ is said to have least upper bound property if every subset $A_0 \subset A$ has a least upper bound. $\mathbb{R}$ has least upper bound property is well known. Now consider the subset $A = ...
0
votes
2answers
88 views

What is the intersection of a countable set of intervals $(n, \infty)$?

If I have a countable set of intervals $\{ A_n \}^{\infty}_{n=1}$ where $A_n = (n, \infty)$, and take the intersection $ \cap_{n=1}^{\infty} A_n $ my assumption is that this set would be ...
1
vote
3answers
27 views

Why when all values x in set A and x is part of set B, A is a subset of B?

Reading an intro to mathematical sets I stumbled on one property that I don't see the logic of: If $\forall{x} \in A \Rightarrow x \in B$, then A is called a ...
1
vote
1answer
144 views

Intersection of Disjoint Unions and deMorgan's Laws for Disjoint Sets

Let $\sqcup$ denote the disjoint union. Then is there nice deMorgan relation we can obtain for: $$\left(\bigsqcup_{i=1}^n E_i\right)^c$$ On the other hand, suppose we have the following ...
-1
votes
1answer
78 views

Question about analysis

We know $x \sim y$ iff $y - x \in \theta \mathbb{Z}$ (mod 1) and $\theta$ irrational defines an equivalence relation on $[0,1]$ with equivalence classes $[x] = \{\{x + n \theta \}\} = Orb (x)$. My ...
1
vote
2answers
138 views

If $f(x,y)=x^2+y$, what is the image of $K=\{(x,y):x^2+y^2\leq 1\}$?

Please disregard the first eight lines of the solution below (which I have provided for completeness; the referenced theorems simply state that continuous functions between metric spaces preserve ...
1
vote
1answer
640 views

Proof for Strong Induction Principle

I am currently studying analysis and I came across the following exercise. Proposotion 2.2.14 Let $m_0$ be a natural number and let $P(m)$ be a property pertaining to an arbitrary natural ...
1
vote
1answer
39 views

Pushforward Filter

While reading some notes I came across the notion of a pushforward pre-filter. If $g:X\rightarrow Y$ is a continuous map of topological spaces and $F$ is a pre-filter on $X$, then then author claims ...
2
votes
1answer
108 views

Proper map on from compact manifolds

I'm stuck on this statement. Could anyone please help me out? Let $X$ be a compact manifold, every map $f: X \longrightarrow Y$ is proper. The definition of proper: a smooth map between manifolds is ...
1
vote
1answer
40 views

On the limit of a Minkowski sum

Consider an open set $\mathcal{O} \subseteq \mathbb{R}^n$. I am wondering if the set $$ \mathcal{S} := \lim_{k \rightarrow \infty} \ \mathcal{O} + \frac{1}{k} \mathbb{B} $$ is open or closed. With ...
0
votes
3answers
136 views

How can I prove that a set of real numbers always have a minimum?

If I take a set of real numbers, say $S$, can I always prove that there is a number $n^*$, contained in the set $S$, which for all other $n$ in that same set, $n^*\leq n$ always holds? I guess this ...
1
vote
1answer
108 views

Example of bijection from $\mathbb{Q} \to \mathbb{Q} \times \mathbb{Q}$

What would be an example of bijection between $\mathbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$. I can think of one: $x \mapsto (x,x+1)$ Does this work? I am not sure.<
3
votes
3answers
445 views

What sets contain $\infty$ and $-\infty$ and why are the Integers closed?

So I'm currently studying from Rudin's Principles of mathematical analysis or colloquially "Baby Rudin" and have stumbled into the second chapter namely basic topology. He lists some sets and states ...
1
vote
1answer
66 views

Limit of constant functions with countable discontinuities

Suppose we have a family of functions $\sum_{i=1}^n \alpha_i 1_{F_i}$ where $1_{F_i}$ is the characteristic function of $F_i \subset \mathbb{R}$, and $F_i$ is countable or $F_i^c$ is countable. This ...
4
votes
1answer
160 views

Explicit bijection between Jordan curves and real numbers

It is my understanding that the set of all Jordan curves and the set of real numbers are of the same cardinality. So, it follows that there should exist a bijection between them. Is there a known, ...