# Tagged Questions

**1**

vote

**2**answers

199 views

### Jacobian of an injective mapping

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. I want to prove two things that seem intuitively right: 1) $f$ is ...

**9**

votes

**1**answer

197 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 ...

**14**

votes

**2**answers

944 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 ...

**2**

votes

**0**answers

124 views

### How many ways we have to prove that a topologically (or analytically) nice mapping is injective?

I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and ...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**12**

votes

**11**answers

1k 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 ...

**2**

votes

**1**answer

226 views

### Finding a good ordering of $\mathbb{Q}$

Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result.
From a research question I am working on I have simplified the ...

**1**

vote

**1**answer

341 views

### Does there exist a countable partition of [0,1] by disjoint closed subsets? [duplicate]

Possible Duplicate:
Why are the integers with the cofinite topology not path-connected?
As in the title, is it possible to find closed, disjoint subsets $C_n$ of $[0,1]$ such that $[0,1] = ...

**2**

votes

**2**answers

322 views

### When a set of measure zero plus itself contains interior

Is there a characterization of measure zero subsets $A$ of $\mathbb R^n$, $n>1$ such that the set $A+A$ contains interior? Here $A+A$ is the set of points $\{ x+y \mid x, y\in A \}$.
Is it true ...

**7**

votes

**3**answers

654 views

### Is the reals the smallest connected ordered topological ring?

The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to ...

**2**

votes

**1**answer

277 views

### showing uniformly continuous

Let $(X,d)$ be a metric space and $(a_n)$ be a sequence of distinct points in $ X$ such that each $a_n$ is a limit point of $X$. If $U_n$ 's are mutually disjoint open neighbourhoods of $a_n$ in $X$. ...

**7**

votes

**1**answer

265 views

### Topological conditions forcing continuity

Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous.
Question: Under what ...

**5**

votes

**2**answers

416 views

### On the uncountability of zero sets

If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain.
I ...

**4**

votes

**0**answers

221 views

### Whitney approximation without second countable

One version of Whitney's approximation theorem states the following:
Let $N$ be a smooth, Hausdorff, second-countable (or paracompact) manifold, then given any continuous function $F:N\to ...

**2**

votes

**1**answer

496 views

### Question about analytic curves

Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...

**63**

votes

**4**answers

8k views

### Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...

**28**

votes

**1**answer

1k 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 ...

**2**

votes

**1**answer

673 views

### A question about measurable structures on function spaces

Hey, I was just wondering, I'm using some of Robert Aumann's ideas about measurable structures on function spaces (From his paper 'Borel structures for Function spaces': ...

**3**

votes

**0**answers

258 views

### For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...

**7**

votes

**3**answers

484 views

### Connectifications?

Like many of my questions, this question is actually aimed at $p$-adic analysis.
One of the main obstacles in doing analysis $p$-adically ist that the $\mathbb{Q}_p$ is totally disconnected.
From ...

**16**

votes

**3**answers

2k views

### Weak and Strong Integration of vector-valued functions

This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...

**6**

votes

**6**answers

1k views

### Uncountable preimage of every point

Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable?
Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the ...

**7**

votes

**2**answers

781 views

### Unusual Space-Filling Curve

Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.
Consider a foliation as a collection of continuous nonintersecting curves that start at (0,0) and end ...

**10**

votes

**2**answers

637 views

### Are there space filling curves for the Hilbert cube ?

There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...

**0**

votes

**2**answers

230 views

### A Jordan Arc in the unit disk

Let D be the open unit disk, and J a Jordan arc (that is a homeomorph of [0, 1]) that lies in D, except J(0) lies on the boundary of D, say J(0)=1. I would like to see that D\J([0, 1]) is a path ...

**1**

vote

**1**answer

1k views

### What is the pure intuition for topological continuity and topology? [closed]

I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity.
The ...

**2**

votes

**3**answers

825 views

### motivation for compactness [duplicate]

Possible Duplicate:
How to understand the concept of compact space
Hello,
I am learning some analysis on my own and
what is the motivation to consider compactness?
eg. I do not understand ...

**3**

votes

**1**answer

574 views

### Name for topology making group action continuous

Fix a set $X$ with right $G$-action. Give $X$ a topology $\tau$ and make $G$ a topological group. (These topologies need not make the action continuous).
We can define another topology $\tau'$ on ...

**6**

votes

**3**answers

507 views

### Locally complete space is topologically equivalent to a complete space

Can someone please tell me where I can find a citeable reference for the following result:
Call the metric space $(X, d)$ "locally complete" if for every $x \in X$ there a neighbourhood of $x$ which ...

**4**

votes

**0**answers

445 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 ...

**6**

votes

**2**answers

425 views

### Can I detect the point of impact without looking at it?

I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...

**0**

votes

**3**answers

205 views

### how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes ...

**1**

vote

**6**answers

2k views

### Cone in a metric space

Hi everybody,
We know the definition of a cone in a Real Banach Space.
I want to know if there is any definition for a cone in an abstract metric space.
Have you ever seen such definition anywhere?
...

**7**

votes

**6**answers

2k views

### When does local invertibility imply invertibility?

Generally, local invertibility does not imply invertibility. However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.
...

**23**

votes

**6**answers

1k views

### Is there a topology on growth rates of functions?

I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions (0,infty) \to (0,\infty), where two functions f_1,f_2 ...

**7**

votes

**1**answer

368 views

### Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...

**14**

votes

**5**answers

955 views

### Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...

**41**

votes

**16**answers

7k views

### Atiyah-Singer index theorem

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...

**3**

votes

**3**answers

657 views

### Does the “continuous locus” of a function have any nice properties?

Suppose f:R→R is a function. Let S={x∈R|f is continuous at x}. Does S have any nice properties?
Here are some observations about what S could be:
S can be any closed set. For a closed set ...