0
votes
1answer
20 views

Any denumerable set is infinite

Currently, I'm learning 'An Introduction to Classical Real Analysis' (Stromberg, 1981) by myself and find that the proof of Theorem (1.55) in pages 29-30 is far beyond my comprehension. Can anybody ...
0
votes
0answers
27 views

comparing cardinality of infinte sets [duplicate]

Let's say I have two infinite sets: 1) the set of all functions from $\mathbb{R}$ to {0,1}; 2) the set of all polynomials whose coefficients are in $\mathbb{R}$. Which is greater? I figure the ...
0
votes
0answers
19 views

Proofs Regarding Open and Closed Sets

I need to prove the following regarding open and closed sets: 1. A set L is closed iff for any converging sequence $(x_n)$ with $x_n\in L$, the limit $x=\lim_{n\to\infty}{x_n}$ is also an element of L ...
2
votes
2answers
22 views

Does there exist a bijective mapping of an open interval with the corresponding closed interval having only finitely many points of discontinuity?

Given $a<b$, is there a bijection $f \colon [a,b] \rightarrow (a,b)$ such that $f$ be continuous except at finitely many points only? I know that there does exist a bijection of $[a,b]$ with ...
-1
votes
1answer
15 views

Symmetric Difference of a Triplet of Sets

I am doing set theory out of interest and am stuck on the following question: Show that: Z \ (X Δ Y) = [Z \ (X ∪ Y)] ∪ ( X ∩ Y ∩ Z) My approach: Beginning with the RHS: [Z \ (X ∪ Y)] ∪ ( X ...
1
vote
2answers
26 views

One one & Onto functions

Are there one one function from the set $A$ to the set $B$? Are there onto function from the set $A$ to the set $B$? Where $A=\{x^2 :0<x<1\}$ and $B=\{x^3:1<x<2\}$.
1
vote
1answer
66 views

Using the Cantor-Bernstein theorem

I'm working through Kolmogorov and Fomin's Introductory Real Analysis text, and I came to a question about showing that some sets have the same power as the continuum. I have seen this question posted ...
0
votes
1answer
31 views

$P=(m,n)$, $Q=(h,j)$. Prove that $P\subseteq Q$ iff $h\leq m\leq n\leq j$

$P=(m,n)$, $Q=(h,j)$. Prove that $P⊆Q$ iff $h≤m≤n≤j$ I have no idea about how to prove it, does anyone could help me? Thanks!
3
votes
1answer
50 views

Is the following a semiring?

I have the following problem: Let $f: X' \rightarrow X$ be any map and $\mathcal{H} \subseteq \mathcal{P}(X)$ a semring. Is $f^{-1}(\mathcal{H})$ a semiring? Thanks for your help!
0
votes
1answer
68 views

Geometrical description of cantor set is uncountable

Why is Cantor set uncountable? I would like to intuitively understand the uncountable nature of Cantor set. When I construct Cantor set I do not feel so.
4
votes
4answers
86 views

Prove that $\text{int(intA)=int(A)}$?

I want to prove that $\text{int(intA)=int(A)}$ (and we are in metric space). I have two questions regarding this. (1). I came up with this proof but don't know if it's correct or not. First I use ...
1
vote
1answer
51 views

Some problems from section 4 of Munkres

I'm right now covering Section 4 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problems in the exercise set after Section 4: Problem 8(c): Show that given $a$ with ...
-1
votes
3answers
41 views

Enumeration of a set

What do this statement mean: For any given $k\in \left[a,b\right]$, let $\left( x_n \right)_{n=1}^{\infty}$ be an enumeration of the set $\left\{c_jy_k^i:i,j\ge 1, \right\}$ where $c_j$ is a real ...
3
votes
1answer
65 views

Lindelof's Covering Theorem

If $A \in \mathbb R^n$ and $F$ be an open covering of $A$. Then there is a countable subcollection of $F$ which also covers $A$. Proof: Let $G=\{A_1,A_2, \cdots\}$ denote the countable collection of ...
0
votes
1answer
33 views

Question on the proof of the following theorem : the image of a compact set under a continuous function is compact

Theorem : Let $f: S \rightarrow T$ be a function from one metric space $(S,d_s)$ to an another metric space $(T,d_t)$. If $f$ is continuous on a compact subset $X$ of $S$, then the image $f(X)$ is a ...
0
votes
3answers
47 views

Cantor's Theorem (surjection vs bijection)

Let me state Cantor's Theorem first: Given any set $A$, these does not exist a function $f:A \rightarrow P(A)$ that is surjective. I understand the proof of this theorem, but I'm wondering why it's ...
0
votes
3answers
35 views

If $S \subseteq T \subseteq M$. Then show that $S$ is compact in metric space $(M,d) \iff S$ is compact in the metric subspace $(T,d)$.

Let $(M,d)$ be an arbitrary metric space and $S,T$ be subsets of $M$. Assume $S \subseteq T \subseteq M$. Then show that $S$ is compact in $(M,d) \iff S$ is compact in the metric subspace $(T,d)$. ...
1
vote
4answers
30 views

Let $X$ be a closed subset of a compact metric space $M$. Then, $X$ is compact.

Theorem : Let $X$ be a closed subset of a compact metric space $M$. Then, $X$ is compact. Query : Since, $M$ is compact, then there is a finite collection $F$ of open sets which covers $M$. Hence, ...
1
vote
1answer
107 views

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets I found this proof on a certain web page A direct proof would be ...
0
votes
3answers
37 views

The set of real numbers and the set of Real valued functions are not similar (equinumerous)

We need to show that the set of real numbers and the set of Real valued functions whose domain is $\mathbb R$ are not similar (equinumerous). Let $\mathbb R$ denote the set of real numbers and $S$ ...
0
votes
1answer
22 views

Define a new operation and prove the field axiom hold for it.

Define $\ a\triangle b=ab$, where $a,b\in\Bbb R^{+}$, the set of positive real numbers. Show that $\exists x \in\Bbb R^{+}$ s.t $a\triangle x=0$. I think the statement is false, because you can not ...
1
vote
0answers
36 views

Existence of extended real number system

How can we prove the existence of Extended real no. such that order and addition and multiplication (which are also relation means set) are also extended from real no. system. Can you please provide ...
0
votes
1answer
34 views

Are any two uncountable sets similar to each other?

Two sets $A$ and $B$ are called similar $\iff$ thee exists a one to one function $F$ whose domain is the set $A$ and whose range is the set $B$. We know that two countably infinite sets should be ...
1
vote
2answers
24 views

Cardinality of a line and a half plane

intuitively it seems like the cardinality of the set of points that make up a line should be different than the cardinality of the set of points that make up a half plane but I couldn't come up with a ...
1
vote
1answer
28 views

How can I form a bijection between these elements?

I am having trouble getting started with this particular problem. Let $A$ be a nonempty set, and let $\mathcal{B}$ be the set of all functions $f:A\to\{0,1\}$. Show that ...
0
votes
3answers
33 views

Show the range of a fuction is (-2,2)

Please help me to solve "show the range of $$h=\frac{-2k}{\sqrt{1+k^{2}}}$$ is $(-2,2)$", thanks! Limit can not be used here!
0
votes
1answer
38 views

Prove the existence of a point not accounted for by mapping from N to R and deduce uncountability of R from this

Let a: $\mathbb{N}\rightarrow\mathbb{R}$ be given. For $a, b \in \mathbb{R}$ such that $a < b$ show that there is a point $c$ in the closed interval $I = [a, b]$ such that $c \notin \{a(n) | n \in ...
0
votes
3answers
85 views

How is the set of all closed intervals countable?

I am trying to figure out the answer to the problem: Show that the set of all closed intervals $[a,b]$ with $a,b \in \mathbb{Q}$ is countable. Now I know that the interval $[0,1)$ for example is ...
0
votes
2answers
101 views

Applications of infinite cardinalities in real analysis

What are some topics in real analysis that make use of infinite cardinalities larger than that of the real numbers themselves, preferably [edit: but not necessarily] topics that are widely applied in ...
1
vote
1answer
40 views

Proof of the uncountability of reals using the diagonal argument—problem?

Consider a common proof of the uncountability of $(0,1]$, as presented here for example: We assume that the reals can be arranged in a sequence $x_k$, represent every number in $x_k$ by its ...
0
votes
1answer
23 views

Cardinality of sets regarding

Consider the following sets of functions on $\mathbb{R}$. $W=$The set of all constant functions on $\mathbb{R}$ $X=$The set of polynomial functions on $\mathbb{R}$ $Y=$ The set of continuous ...
1
vote
1answer
35 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
0
votes
2answers
27 views

Proving a set identity

If $A \subseteq X$ and $A_{\alpha}$ is a collection of all such subsets, I need to prove that: $$\left(\bigcap_{\text{all }\alpha} A_\alpha\right)^c = \bigcup_{\text{all }\alpha} A_{\alpha}^c$$ My ...
1
vote
3answers
44 views

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$?

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$ ? I have tried to construct this set from countably union ...
1
vote
2answers
48 views

A nested cover of a set that eventually has infinite intersection with every infinite subset

Prove the following Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of sets with $$A_1 \subset A_2 \subset A_3 \dots $$ and $B \subset \bigcup_{n = 1}^{\infty} A_n$. If for every infinite subset $E$ ...
2
votes
3answers
67 views

Show that if $f$ is injective, then $f^{-1}(f(C))=C$ [duplicate]

Once again I am stuck on a question from Lay's Introduction to Analysis with Proof: Suppose that $f : A \rightarrow B$ and let C $\subseteq$ A and $D \subseteq B$. Show that if $f$ is injective, ...
1
vote
1answer
22 views

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions.

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions $f: X \rightarrow \mathbb R$ in each of the ...
2
votes
0answers
26 views

function over countable union, intersection, etc

is it true that for a give function, or a linear transformation, $L$, in $\mathbb{R}^n$, if $S=\bigcup_{j=1}^\infty T_j$, then $L(S)=L(\bigcup_{j=1}^\infty T_j)=\bigcup_{j=1}^\infty L(T_j)$? What if ...
0
votes
0answers
81 views

Proof of the cardinality of continuous functions from $[0,1]$ to $[0,1]$.

I've been thinking about the cardinality of continuous functions from $[0,1]$ to $[0,1]$. I know that the cardinality is the same as that of $[0,1]$ and the standard proof using the fact that such a ...
1
vote
0answers
19 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
1
vote
1answer
40 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
-2
votes
2answers
103 views

How do i construct $C^\infty$?

I'm trying to define $C^\infty$ rigorously and i have a trouble with this. Mathematical Induction should be used, but i dunno where to apply this. I'm going to illustrate what i tried below: Before I ...
5
votes
0answers
107 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
1
vote
2answers
46 views

Construction of a small but fat set? [duplicate]

Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?
0
votes
1answer
48 views

Is $\sup\{t>0:F(t)>0 \in [0,t]\}$ an incorrect math expression?

I saw the following in a journal paper and the notation looks wrong - am I right? $$t_1 = \sup\{t>0:F(t)>0 \in [0,t]\}$$ I would like to translate this into an English sentence, but I don't ...
3
votes
4answers
115 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). ...
1
vote
0answers
26 views

problem with my construction of sequence of sets

This is an exercise from a real analysis book. It has many parts. Assume a) and b) are true: a) Suppose $F$ is closed and $O$ is open subset of $\mathcal{R}$ and $F \subset O$, then there is a ...
2
votes
2answers
66 views

Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$ f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases} $$ Let $y \in \mathbb{R}$. How would I prove that there ...
1
vote
0answers
20 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
1
vote
1answer
88 views

Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...