All Questions
Tagged with gn.general-topology real-analysis
247 questions
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, $f(S)\subset Y$ ...
- 39k
107
votes
9
answers
36k
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 ...
- 41.4k
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 ...
- 629
61
votes
1
answer
5k
views
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...
- 32.5k
38
votes
13
answers
5k
views
Continuous relations?
What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?
I ...
- 862
31
votes
13
answers
6k
views
Classic applications of Baire category theorem
I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 407
26
votes
2
answers
5k
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:
...
- 29.8k
23
votes
3
answers
4k
views
Continuous functions taking uncountably many values countably often
Let $f$ be a continuous function defined on the closed interval $[0,1]$. Clearly $f$ is bounded and attains its bounds.
Then my question is how often can $f$ take a value in its range countably many ...
- 4,862
23
votes
4
answers
2k
views
Which is the correct ring of functions for a topological space?
There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...
- 54.6k
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 ...
- 6,741
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 ...
- 1,014
19
votes
3
answers
1k
views
Is there a Cantor set $C$ in $\mathbb{R}^{2}$ so the graph of every continuous function $[0,1]\rightarrow [0,1]$ intersects $C$?
Consider the Cantor ternary set on the real line with the usual topology and define a Cantor set to be any topological space $C$ homeomorphic to the Cantor ternary set.
The idea is to construct a ...
- 2,136
19
votes
1
answer
556
views
Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?
Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$?
Remark 1. The answer to the ...
- 17.7k
18
votes
2
answers
2k
views
Generalization of Darboux's Theorem
Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let $U\...
- 2,933
18
votes
2
answers
630
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 ...
- 32.1k
18
votes
2
answers
1k
views
Comparing "axiomatized function spaces"
This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing ...
- 21.2k
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
- 831
17
votes
2
answers
905
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 A\...
- 3,584
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?
- 41.8k
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 ...
- 32.1k
16
votes
2
answers
1k
views
Is there always a way up?
I am trying to find a simple criterion for a real continuous function $f$ on a connected, open subset $U$ of $\mathbb R^n$ that would imply the following property (P)
For any $x, y \in U$ such that $f(...
- 1,608
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 536
15
votes
1
answer
904
views
Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?
Willie Wong asked here (MO) and here (MSE) very interesting question.
As he phrased it:
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ ...
- 225
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 $\mathbb R^\omega$ (...
- 41.8k
14
votes
3
answers
547
views
Recognizing Lipschitz functions up to change of target metric
Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.
Is ...
- 143
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 ...
- 1,396
12
votes
2
answers
607
views
Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set
It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary set?...
user36763
12
votes
2
answers
678
views
Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
- 779
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 ...
- 221
11
votes
3
answers
1k
views
Can there be two continuous real-valued functions such that at least one has rational values for all x?
Of course, no continuous real valued non-constant function can attain only rational or irrational values, but can there be a pair of nowhere-constant continuous functions f and g such that for all x, ...
- 656
11
votes
3
answers
890
views
Structure theorems for compact sets of rationals
Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
- 1,882
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^...
- 2,933
11
votes
1
answer
704
views
Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$
We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions.
One can generalize the definition above by taking pointwise limit of ...
- 623
11
votes
0
answers
615
views
Is every Baire metric space a complete metric space in disguise?
I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...
- 61.9k
11
votes
0
answers
322
views
Does any real function have a Lipschitzian restriction on $D$?
Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
- 4,074
10
votes
2
answers
3k
views
Continuous function from $[0,1]$ to $[0,1]$
Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
- 1,001
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 ...
- 1,959
10
votes
1
answer
571
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 299
10
votes
1
answer
2k
views
Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
10
votes
1
answer
394
views
Maximally continuous extension of continuous functions from $\mathbb Q$ to $\mathbb R$
Let $f: \mathbb Q \to \mathbb R$ be a continuous function.
An extension of $f$ is a function $\tilde f: \mathbb R \to \mathbb R$ such that $\tilde f = f$ on $\mathbb Q$.
We say an extension $\tilde f$ ...
- 6,205
9
votes
5
answers
2k
views
Homeomorphism of the rationals
In working with the classification of stable vector bundles on $\mathbb{P}^2$, I've found that I need to answer a fairly basic question from analysis/point set topology. Here it is.
Suppose $f:\...
- 5,857
9
votes
1
answer
224
views
Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
- 19.6k
9
votes
2
answers
424
views
Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
- 193
9
votes
1
answer
918
views
A Besicovitch-type Covering Theorem
In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
- 1,245
9
votes
2
answers
466
views
Small uncountable cardinals related to $\sigma$-continuity
A function $f:X\to Y$ is defined to be
$\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
- 41.8k
9
votes
1
answer
831
views
Baire category theorem for uncountable unions
Any compact Hausdorff space $X$ is a Baire space:
if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets,
also known as a set of first category),
then $X$ is empty.
I am ...
- 37.8k
9
votes
1
answer
428
views
The cardinality of projections of subsets of the Hilbert cube by inner products
I have three related questions.
Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
- 3,104
9
votes
1
answer
339
views
A topological characterisation of a.e. continuity
We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
- 6,205
9
votes
0
answers
569
views
A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 41.8k