1
vote
2answers
194 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
1answer
195 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
2answers
939 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
0answers
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
2answers
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
11answers
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
1answer
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
1answer
326 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
2answers
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
3answers
623 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
1answer
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
1answer
261 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
2answers
414 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
0answers
220 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
1answer
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 ...
60
votes
4answers
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
1answer
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
1answer
670 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
0answers
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
3answers
483 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
3answers
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
6answers
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
2answers
779 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
2answers
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
2answers
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
1answer
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
3answers
806 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
1answer
572 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
3answers
505 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
0answers
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
2answers
424 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
3answers
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
6answers
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
6answers
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
6answers
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
1answer
364 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
5answers
949 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
16answers
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
3answers
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 ...