0
votes
0answers
16 views

Definition of a Paracompact space

I have a question about the definition of a paracompact space. We said that a space $X$ is paracompact iff $X$ is $T_2$ and if any open covering of $X$ has a finer locally-finite covering. I don't get ...
1
vote
1answer
19 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
0
votes
1answer
29 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
0
votes
1answer
27 views

If $S$ is dense in $L^{2}$. Is it true that $pS=\{pf| f\in S, pf\in C^{\infty}\} $ is dense?

Let $S=\{f\}$ be a set of function defined in a compact subset $\Omega\subset \mathbb{R}^{n}$ such that $S$ is dense in $L^{2}(\Omega)$. Is it true that for $p\neq 0$ a rational function $pS=\{pf| ...
5
votes
1answer
60 views

Visualisation of the smash product

wedge product, join etc. all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no ...
2
votes
1answer
42 views

Proof of a distance

I have one distance shown as an example in a book but I'm striving to demonstrate that it is effectively a distance. here it is said : let $U=\{z\in\mathbb{C, |z|=1}\}$ we can get a distance on $U$ ...
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
-2
votes
0answers
33 views

Does an lp norm induce a ball topology? [closed]

Namely, does the metric $$||x - y||_p$$ induce the usual ball topology that a metric induces? I wasn't able to find any results regarding this on a quick Google search.
1
vote
0answers
26 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
0
votes
1answer
55 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
2
votes
0answers
46 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
3answers
72 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
0
votes
1answer
32 views

Proof of an open set or closed set

I'm struggling on a proof that I can't proof correctly. Let $A=\mathbb{Z}$, $B=\{n-\frac{1}{2n} | n \in \mathbb{N}*\}$ I could prove easily that A is a closed set and B as well : $\overline A =$ ...
1
vote
2answers
84 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
1
vote
1answer
26 views

Prove that the a modified Cantor Set is not Jordan-Measurable

Let $C_0 = [0,1]$ and if $C_n$ is given as a disjoint union of intervals, construct $C_{n+1}$ by removing from each interval $I$ an open interval of length $(n+2)^{-2}|I|$ in the middle of each ...
0
votes
2answers
47 views

If $a$ is a limit point of $f^{-1}(b)$, then the linear mapping $f'(a)$ is not injective.

Let $U\subset\mathbb{R}^m$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function. Suppose that there exists $b\in\mathbb{R}^n$ and $a\in U$ such that $a$ is an accumulation point of ...
0
votes
1answer
29 views

Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
3
votes
1answer
81 views

What are the $n$th roots of the identity function?

What are all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f^n=I$ where $f^n$ denotes the composition $f\circ f\circ f\dots \circ f$ of $f$ with itself $n$ times, and ...
3
votes
1answer
80 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
0
votes
2answers
70 views

How can I make this proof more precise / detect the small error

For a presentation seminar I had to give a mathetmatical presentation, and amongst others I explained the following lemma with accompanying proof: Let $G:I \to \mathbb{R}$ be continuous, where $I$ is ...
0
votes
3answers
65 views

Example of a function?

$f$ is a discontinuous and bounded function defined on a closed set $C$. Also there exists a non-discrete closed subset in the image of $f$ such that it's inverse is open. Can you give an example ...
1
vote
2answers
73 views

On the abstract bootstrap principle in the book “Nonlinear Dispersive Equations” by Terence Tao

In Terence Tao's book "Nonlinear Dispersive Equations", he gives the following "Abstract bootstrap priciple": "Let $I$ be a time interval, and for each $t \in I$ suppose we have two statements, a ...
0
votes
1answer
36 views

Derivatives and connected sets

I'd like to prove the following two propositions: a) Derivatives maps connected sets into connected sets b) There exist functions which maps connected sets into connected sets that are not the ...
1
vote
2answers
71 views

Fundamental groups and some properties

I have some basic questions about fundamental groups that came up when I tried to prove a few things: I am sorry that they are kind of informal questions, but I could not find any answers to them in ...
2
votes
1answer
56 views

Basic Analysis Help. Open & Closed Sets; Topology

I need some help! First, let me say that I am Mathematics major, as I'm a senior in college finishing up my undergraduate work with the hope of going to graduate school for Mathematics in the future; ...
0
votes
2answers
55 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
0
votes
1answer
46 views

Questions about van Kampen's theorem.

I just read some things about van Kampen's theorem that threat this one from a different perspective than we discussed in class and this brought up a few questions: It was said that the images of the ...
2
votes
2answers
50 views

About Path-connected

Let $a=(a_1,a_2,...,a_k)$ and $b=(b_1,b_2,...,b_k)$ be points in k-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0,1]$ with values in ...
1
vote
1answer
61 views

Weak* compactness of the unit ball

Things that we know: In any topological space compactness implies sequential compactness If E is any topological space the then the closed unit ball $$ B_E=\{f\in E^*; \|f\|\leq 1\} $$ is compact ...
3
votes
1answer
61 views

Question about Hatcher's book CW complex

I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood ...
-1
votes
1answer
24 views

Regularity of Boundaries

From my understanding, regularity of boundaries are effectively talking about the continuity of the boundary of a set. For example, if I consider $\Omega \in \mathbb{R}^2$, where $\Omega$ is the unit ...
1
vote
1answer
35 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
0
votes
1answer
17 views

$X$ $T_4$, then $X/R$ $T_4$?

I found out that if $X$ is $T_4$ and $R$ is a closed equivalence relation, then $X/R$ is also $T_4$. I was just wondering whether the same is true for $R$ being an open equivalence relation?
10
votes
1answer
239 views

What is the Atiyah-Singer index theorem about?

I was just a little bit curious about the general statement of this theorem. Honestly, I am not at all interested in fully understanding this, so it is not that I am too lazy to read plenty of books ...
10
votes
1answer
100 views

Topologists glueing, cutting and so on. When is this rigorous?

I often see that things in topology are explained very non-rigorously recently. Thereby I mean that it is said that we can cut something and glue something together and so on in order to identify two ...
0
votes
0answers
53 views

$E$ is closed $\leftrightarrow E^{c}$ is open

I'm having difficulty following this proof provided in Principles of Math. Analysis by Rudin. Pf First suppose $E^{c}$ is closed. Choose $x \in E$. Then $x \notin E^{c}$, and $x$ is not a limit ...
0
votes
1answer
22 views

Difference of infima exceeds infima of difference

In a project I am working on I encountered the following statement. Let $X$ be a compact metric space and $f,g:X\rightarrow\mathbb{R}$ two upper semicontinuous functions. Then $$ \inf_{x\in ...
1
vote
1answer
52 views

Is this a non empty perfect set with no rational numbers?

Let $A$ be the Cantor set and $B = \pi\, (-A \cup A)$. Is B a perfect set with no rational numbers?
1
vote
0answers
37 views

Classic applications of Baire category theorem [duplicate]

Problem: Suppose that $f:\Re^+\to\Re^+$ is a continuous function with the following property: for all $x\in\Re^+$, the sequence $f(x), f(2x),f(3x)\cdots$ tends to $0$. Prove that ...
0
votes
1answer
45 views

Dense Cantor set approximation

I am reading Measure, Topology and Fractal Geometry by Edgar and in the first few pages he defines the Cantor set and a dense approximation to the Cantor set. He says that $\frac{1}{4} = ...
0
votes
0answers
45 views

Complex analysis winding number

We have that $f:\mathbb S^1 \rightarrow \mathbb S^1 $ and $f(z)=f(1)\widehat{\phi}(z)$ with $\widehat{\phi}(\exp{2\pi it}) = \exp(2 \pi i \phi(t)),$ where $\phi:I \rightarrow \mathbb{R} $ is a ...
2
votes
0answers
28 views

CW structure on $P^n \mathbb{R}$(projective space)

I am supposed to show that $P^n \mathbb{R} = e_0 \cup\cdots\cup e_n$, where $e_i$ is an $i$-dimensional cell. I also know that there is a quotient map $S^n \rightarrow S^n/\tilde - = P^{n-1} ...
1
vote
1answer
64 views

Why $(X,d)$ is a complete $\mathbb{R}$-tree?

Definition. An $\mathbb{R}$-tree is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$) joining each pair of points $x,y \in X$; if $[x,y] \cap [y,z] = ...
2
votes
1answer
70 views

What is a CW complex

In a lecture, I have written down the following definition for CW complexes. $X= \bigcup_i \{e_i\}$ and the $\{e_i\}$ form a partition. Furthermore $e_i$ is homeomorphic to $B(x,1)\subset ...
1
vote
1answer
55 views

Show that topologies are the same

I just read a proof where it was said that if for each element in the topology 2 we find an element in topology 1 that is contained in this set and vice versa, then they are the same. How do I see ...
0
votes
0answers
34 views

Is the closure of the cells in a CW complex compact?

As the cells in a CW complex afaik are homeomorphic to the open ball by definition, I was wondering whether this also means that their closure is a compact set? And if this is true, I would be ...
0
votes
0answers
31 views

CW complex in $\mathbb{R}^n$?

I am supposed to show that a finite CW complex is homeomorphic to a compact subset of some $\mathbb{R}^n$, where $n$ is large enough. My idea was the following: A CW complex is also T3, which means ...
0
votes
0answers
24 views

Locally compact CW complex

I want to prove the folowing: A CW complex is locally compact iff every point has a neighbourhood that intersects with just finitely many cells. I already did the implication "locally compact", ...
0
votes
1answer
69 views
1
vote
0answers
12 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...