# Tagged Questions

**0**

votes

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39 views

### Help on step of “The real numbers are uncountable” proof [migrated]

I'm currently a student in a Real Analysis class, and we today we discussed a proof of why the real numbers are uncountable--however, I'm having a little trouble understanding why our proof is true. ...

**7**

votes

**2**answers

212 views

### Cofinality of a $\sigma$-ideal of $\mathbb{R}$

The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...

**13**

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**0**answers

291 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is motivated ...

**14**

votes

**2**answers

445 views

### Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...

**16**

votes

**2**answers

455 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

**27**

votes

**2**answers

843 views

### Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent

Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?

**10**

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**2**answers

365 views

### Is sigma-additivity of Lebesgue measure deducible from ZF?

Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. ...

**16**

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**2**answers

492 views

### Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...

**5**

votes

**1**answer

361 views

### Does a nonlinear additive function on R imply a Hamel basis of R?

A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...

**4**

votes

**1**answer

318 views

### Must the Minkowski sum of a Borel set and a *closed* ball be Borel?

Let A be a Borel set in R^n. Must then A + B(0,1) be Borel?
Here B(0,1) is the closed ball centered at 0 of radius 1.
I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a ...

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votes

**1**answer

734 views

### Is it necessary to use AC to solve this problem ?

Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, ...

**2**

votes

**3**answers

259 views

### Construct a fixed-point set operator

How to find an uncountable set $S$, and construct an function $f : 2^S
\longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in
T$?
for example, let $S =\mathbb{R}$, how can I ...

**5**

votes

**1**answer

288 views

### Arbitrary small positive lower semi continuous functions

This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...

**4**

votes

**2**answers

521 views

### Finite measure on the power set

Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...

**18**

votes

**4**answers

1k views

### Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?

One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...

**5**

votes

**3**answers

893 views

### Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...

**4**

votes

**2**answers

292 views

### Heights of several interesting posets

Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$).
Define several sets of total functions, in each ...

**22**

votes

**2**answers

689 views

### Codimension of Measurable Sets

I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...

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**2**answers

1k views

### Why is there no Borel function mapping every countable set of reals outside itself?

A choice function maps every set (in its domain) to an element of itself. This question concerns existence of an anti-choice function defined on the family of countable sets of reals. In an answer to ...

**40**

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**13**answers

4k views

### Proofs of the uncountability of the reals.

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see a variant proof (as it did not use the diagonal ...

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vote

**0**answers

370 views

### Is there a Real valued function with image of every open interval the whole real line [duplicate]

Possible Duplicate:
Function with range equal to whole reals on every open set
Hello,
My problem is the following
"Is there a Real valued function with image of every open interval the whole ...

**15**

votes

**1**answer

1k views

### Does Arzelà-Ascoli require choice?

Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...

**53**

votes

**5**answers

3k views

### Does pointwise convergence imply uniform convergence on a large subset?

Suppose $f_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero.
Is there an uncountable subset $A$ of $[0,1]$ so that $f_n$ converges uniformly on $A$?
Is there a ...

**3**

votes

**3**answers

599 views

### Monotone injection of an ordinal into $[0,1]$

This is related to my recent question and would provide a natural positive answer to Question 2. I am sure this must be known to experts.
Question: Is there a monotone injection $(\omega_1,<) ...

**5**

votes

**3**answers

2k views

### Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...

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votes

**2**answers

601 views

### Partition of R into midpoint convex sets

We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$.
My question is: is it possible to partition $\mathbb{R}$ ...

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votes

**0**answers

440 views

### continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...

**59**

votes

**9**answers

14k views

### solving f(f(x))=g(x)

This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...

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votes

**5**answers

971 views

### Cardinality of Equivalence Classes of Cauchy Sequences

What's the cardinality of a single equivalence class of Cauchy sequences in ℚ?
To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...