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Question regarding the image of a polynomial map containing a small box

I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously. Let $\delta, \varepsilon > 0$. Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
Johnny T.'s user avatar
  • 3,625
0 votes
3 answers
554 views

Converting a bounded metric into an unbounded metric

Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\...
JohnA's user avatar
  • 710
10 votes
3 answers
414 views

Is an open subset of a rigid space rigid?

Let $X$ be a locally compact Hausdorff space. Call $X$ rigid if its only autohomeomorphism is the identity, $\operatorname{Homeo}(X)=\{1\}$. Questions: Let $X$ be rigid. Is it true that every open ...
Bedovlat's user avatar
  • 1,959
15 votes
0 answers
409 views

Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?

Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
Taras Banakh's user avatar
  • 41.9k
17 votes
1 answer
794 views

Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?

Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
Taras Banakh's user avatar
  • 41.9k
2 votes
0 answers
65 views

Splitting of ordinals of oscillation ranks of a Baire $1$ function

Denny and Tang proved that Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$ Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
Idonknow's user avatar
  • 623
7 votes
1 answer
374 views

Is each $G_\delta$-measurable map $\sigma$-continuous?

Definition. A function $f:X\to Y$ between topological spaces is called $\bullet$ $G_\delta$-measurable if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G_\delta$ in $X$; $\...
Taras Banakh's user avatar
  • 41.9k
17 votes
1 answer
988 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
Mohammad Golshani's user avatar
3 votes
0 answers
92 views

Arithmetic progressions inside non meager sets

If $A \subseteq \mathbb{R}$ is non-meager Borel set, then $A$ contains arithmetic progressions of every finite length. I know that this is false if we do not assume that $A$ is Borel. In particular, ...
George's user avatar
  • 31
1 vote
1 answer
245 views

Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?

Let $X$ be a metric space. In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
Idonknow's user avatar
  • 623
2 votes
0 answers
192 views

Generalize upper semicontinuous regularization using Borel Hierachy

Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$. ...
Idonknow's user avatar
  • 623
1 vote
1 answer
162 views

Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?

Ian Morris quoted the following: For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
Idonknow's user avatar
  • 623
4 votes
1 answer
470 views

Covering measure one sets by closed null sets

(The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.) For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval $[0,1]$, define $$\newcommand{\card}[1]{\...
Piotr Szewczak's user avatar
7 votes
3 answers
369 views

Does a certain contractive mapping have a fixed point?

Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying $$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$ where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...
Isra El's user avatar
  • 169
2 votes
1 answer
336 views

Separability of $L^1$ in $L^2$ topology

In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls $$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$ Is $L^1(0,1)$ separable in this topology?
hye's user avatar
  • 23
13 votes
3 answers
820 views

Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?

This question is related to another one that I asked two days ago. Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with the following two properties? The ...
Transcendental's user avatar
2 votes
0 answers
279 views

Can a bounded open set in $R^n$ be always approximated from outside with a finite union of dyadic cubes?

Suppose we have a bounded open set $S$ in $R^n$. Consider the collection of closed dyadic cubes $C_k$'s (https://en.wikipedia.org/wiki/Dyadic_cubes). I was wondering if there always exists a finite ...
KPU's user avatar
  • 131
1 vote
1 answer
118 views

Almost periodic function and closed spaces

We denote $X_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X_{2\pi}+X_{3\pi}.$$ I think that $X_{2\pi}+...
Flo140's user avatar
  • 75
4 votes
1 answer
222 views

Is every regular Borel outer measure topologically additive?

If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive? If so what is a proof or a counterexample? Definitions: Topologically Additive: $X$ is a topological space, $m$ ...
fruitninja's user avatar
2 votes
1 answer
265 views

characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
Kasper Cools's user avatar
-3 votes
2 answers
7k views

Continuous map from $\mathbb R^2$ to $\mathbb R$? [closed]

There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$? I guess what I mean is ...
wurthless_nurd's user avatar
5 votes
1 answer
654 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
Bumblebee's user avatar
  • 1,093
1 vote
1 answer
604 views

Partition of Real Number [closed]

Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
W.Smith's user avatar
  • 275
155 votes
4 answers
18k views

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
Willie Wong's user avatar
0 votes
1 answer
55 views

On 1-iso maps and subsets of the unit circle

Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
T.KM's user avatar
  • 97
0 votes
1 answer
843 views

$C^{\infty}_{loc}$-convergence - right definition

Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
Ben's user avatar
  • 35
-3 votes
1 answer
230 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
yuta's user avatar
  • 3
2 votes
2 answers
762 views

Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism?

I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of $\...
AnotherPerson's user avatar
12 votes
3 answers
440 views

Is a certain subset of the disc a convex set?

Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that? Draw a Cantor set $C$ on the circle ...
mahdi mz's user avatar
  • 221
0 votes
1 answer
482 views

Complement of a finite union of convex sets

Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components. I ...
smyrlis's user avatar
  • 2,933
21 votes
3 answers
610 views

Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
Trevor J Richards's user avatar
6 votes
1 answer
188 views

On continuous perturbations of functions of the first Baire class on the Cantor set

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
Taras Banakh's user avatar
  • 41.9k
7 votes
0 answers
227 views

Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
Taras Banakh's user avatar
  • 41.9k
-1 votes
1 answer
346 views

An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
robert caro's user avatar
1 vote
0 answers
178 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\...
user223935's user avatar
2 votes
0 answers
355 views

Existence of topology on the space of continuous functions

Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...
CodeGolf's user avatar
  • 1,835
4 votes
2 answers
256 views

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \...
jack's user avatar
  • 3,153
7 votes
1 answer
798 views

Intersection of connected components in $\mathbb{R}^n$

Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^* \in \mathbb{R}^n\setminus K$. Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains $x^*$....
Dominic van der Zypen's user avatar
3 votes
4 answers
934 views

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...
Mirko's user avatar
  • 1,375
68 votes
2 answers
2k views

Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
Abcd's user avatar
  • 629
1 vote
0 answers
99 views

Set nor its compliment contain an uncountable closed set [closed]

Does there exist a set $X$ subset of the real numbers such that no uncountable closed set is contained in $X$ or $X^c$?
Leader47's user avatar
  • 121
11 votes
4 answers
2k views

Inserting an open and simply-connected set between a compact set and an open set

In a paper I am reading, the following is considered obvious: Let $K$ be a compact and connected subset of $\,\mathbb R^2$, with $\mathbb R^2\smallsetminus K$ also connected, and $U\subset \mathbb R^...
smyrlis's user avatar
  • 2,933
1 vote
0 answers
260 views

Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
Mark's user avatar
  • 11
2 votes
0 answers
206 views

Regularity of Dirac measure on Baire sets [closed]

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
Richard Hevener's user avatar
1 vote
0 answers
525 views

Separability of the space $C(C[0, 1], \mathbb{R})$

Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$. I am wondering that ...
gregarki khayal's user avatar
23 votes
1 answer
706 views

Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...
Asaf Shachar's user avatar
  • 6,741
2 votes
0 answers
343 views

continuity with respect to weak-${\ast}$ topology

Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
CodeGolf's user avatar
  • 1,835
2 votes
1 answer
800 views

A question about Skorokhod metric

I have a question related to the Skorokhod distance. Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
CodeGolf's user avatar
  • 1,835
2 votes
1 answer
135 views

Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$

Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g. http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g Now define ...
CodeGolf's user avatar
  • 1,835
7 votes
1 answer
2k views

If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?

Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
Silvio Levy's user avatar