1
vote
0answers
37 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
3
votes
0answers
51 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
0
votes
1answer
63 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ of ...
0
votes
2answers
44 views

Why is this a quotient map

Is there a direct way to see that $p \times id : [0,1]^2 \rightarrow S^1 \times [0,1]$ is a quotient map with $(p \times id)(x,y) = (e^{ix},y)$? By direct way, I mean is there an obvious argument why ...
0
votes
3answers
48 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
3
votes
1answer
52 views

Disconnected Topoological Space with Intermediate Value Property

Does There exist a disconnected topological space with intermediate value property? Intermediate Value Property states that 'a topological space X is said to have intermediate value property if for ...
0
votes
2answers
21 views

Nullhomotopic map extended

I have troubles understanding this proof: Let $h:S^1 \rightarrow X$ be a continuous map, then we have that if $h$ is nullhomotopic, $h$ can be extended to a continuous map $k:B^2 \rightarrow X.$ ...
8
votes
4answers
1k views

Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
0
votes
1answer
31 views

Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
-1
votes
1answer
60 views

Topological properties of $(0,1)\times \{0\}$

I am having a real hard time solving simple proofs involving open sets. I am confronted with this one: Is $(0,1)\times \{0\}$ open? Is it compact? What is its interior? I know $(0,1)$ is open. ...
2
votes
0answers
30 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
1
vote
1answer
38 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
1
vote
3answers
94 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
6
votes
1answer
70 views

Show that there is sequence of homeomorphism polynomials on [0,1] that converge uniformly to homeomorphism

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...
-1
votes
1answer
68 views

Topology question about open spaces of a topological space homeomorphic to the full set. [closed]

Let $\mathcal{U}$ be an open subset of $\mathbb{R}^m$ such that there is homeomorphic $f$ from $\mathcal{U}$ to $\mathbb{R}^m$ and also $f$ is an uniformly continous function. Show that ...
0
votes
1answer
28 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
0
votes
0answers
22 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
24 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
33 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
66 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
43 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
42 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 ...
1
vote
0answers
27 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
56 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 $B$ ?
1
vote
3answers
95 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
33 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
3answers
108 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
28 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
50 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
30 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
83 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
83 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
2answers
66 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
74 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
77 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
57 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
82 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
47 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
52 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
67 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
64 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
25 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
39 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
261 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 ...