Questions tagged [continuous-homomorphisms]

To be used with questions about continuous homomorphisms into a space carrying both a topological and an algebraic structure.

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Continuity of adjoint of algebra homomorphism

Consider unital (complex) abelian Banach algebras $A$ and $B$ with corresponding ideal spaces $\Sigma_A$ and $\Sigma_B$. Suppose $\rho:A\to B$ is a continuous algebra homomorphism that maps $1_A$ to $...
Oskar Vavtar's user avatar
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A continuous injective group homomorphism $GL_n(\mathbb{R}) \to O(n)$

Is there a continuous injective group homomorphism $GL_n(\mathbb{R}) \to O(n)$? I'm struggling to construct one - given a matrix $A$ in $GL_n(\mathbb{R})$, how can we construct a matrix $O(n)$ in ...
Robin's user avatar
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What is the relationship between the kernel of a homomorphism and its degree as a covering map?

Let $\phi: A \rightarrow B$ be a continuous homomorphism between Lie groups (or any appropriate generalization). Is there any relationship between the degree of $\phi$ as a covering map of and the ...
CBBAM's user avatar
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On the size of a certain topology defined by an inclusion function.

I found this problem in chapter 7 of biglist. This is supposed to be an easy problem, but I have no idea how to approach it. The problem goes as follows: Let $(X, \mathcal{T})$ be a topological space ...
Ryukendo Dey's user avatar
1 vote
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Proving that a bijective function from unit circle to unit square is continuous

I was asked to prove that the unit circle is homeomorphic to the unit square(both closed), so I came up with this function: $$f:\overline{B}_{1}^{\left\Vert \cdot\right\Vert _{2}}\rightarrow\overline{...
Abzikro's user avatar
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Why aren't all surjective continuous homomorphisms of topological groups open?

My apologies if this a naive question. Let $\phi:G\rightarrow H$ be a continuous surjective homomorphism of topological groups. Denote the kernel of $\phi$ by $N$, then: $$\begin{align} f:G\times N&...
Chris's user avatar
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Why does $\phi_*[X,Y]=[\phi_*X,\phi_*Y]$? [duplicate]

I am a high schooler self-studying differential geometry. I am stuck trying to show the following: Let $\phi:M\rightarrow N$ be a diffeomorphism, and $X$ and $Y$ are vector fields on $M$. Show that $\...
moboDawn_φ's user avatar
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Direct Sum of Topological $R$-modules

Please help me. I am proving the following property: Let $R$ be a topological ring and $\{A_{\lambda}\}_{\lambda\in\Lambda}$ be a family of topological $R$-modules. Let $B_{\lambda}$ be an ...
YSA's user avatar
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Existence of a continuous mapping. [closed]

Do there exists a continuous function $f: \mathbb{R} \to \mathbb{R}_{+}$ such that $$ f(-\log_x y) = x^{-y},$$ for all $x \in \mathbb{R}_{+}$ and $y \in \mathbb{R}_{+}$ ?
Debora Ozassa's user avatar
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Are all countable sortable sets homeomorphic to $\mathbb{N}$?

In any ordered metric space, let a set $S$ be sortable if it can be placed into 1-1 correspondence with $\mathbb{N}$ such that $x > y \iff n(x) > n(y)$ for all $x,y \in S$. I conjecture the ...
SRobertJames's user avatar
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Does a contradiction occurs about a homomorphism on $C(\Omega)$?

In the page 601 of 'HOMOMORPHISMS OF BANACH ALGEBRAS'(Bade and Curtis) https://www.jstor.org/stable/pdf/2372972.pdf, we get an inequality \begin{align*} ||\nu(x_m - x_n)|| \geq \rho_{\mu(C(\Omega))}(\...
Yushiro Aoki's user avatar
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143 views

Group homomorphism between multiplicative groups of rings

My concrete example is that the determinant is a group homomorphism between $GL(n, \mathbb{C})$ and the non-zero complex numbers. But we know that $GL(n, \mathbb{C}) = M(n, \mathbb{C})^{\times}$ is ...
Jagerber48's user avatar
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Coordinate ball union a point in its boundary

Let $M$ be a $n$-dimensional topological manifold, and $U \subseteq M$ be an open set such that there is a homeomorphism $\phi: U \to B$ where $B$ is the $n$-dimensional open unit ball. Question: Let $...
Zhang Yuhan's user avatar
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Continuous functions between p-adic numbers [closed]

Let $p\neq q$ two prime numbers. Are there any interesting continuous functions $\mathbb Q_p\to \mathbb Q_q$ or $\mathbb Z_p \to\mathbb Z_q$? Or more specifically how big is the set $C( A,B)$ (...
Luis Antonio Sanchez's user avatar
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Continuous homomorphisms on the circle

I have the next problem: Let $G$ be a compact abelian group and $H$ the group of continuous characters on G (i.e. continuous homomorphism from G to the circle $T$) with the topology of pointwise ...
Odi's user avatar
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Unitary characters of $1$-torus

This may be really elementary but I'm lost. Let $S^1\subseteq\mathbb{C}$ be the unit circle, which is a group with complex multiplication. Let $\mathbb{T}:=\frac {\mathbb{R}}{\mathbb{Z}}$ be the $1$-...
AlienRem's user avatar
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Is this function a homeomorphism for two rectangular regions in the Moore plane?

Let $X=[0,1]\times [0,1]$ and $Y=[2,3]\times [0,1]$ be a subspaces from Moore plane. Define function $f,g,h\colon X\to Y$ by $f(x,y)=(x+2,y^2)$, $g(x,y)=(x+2,1-y)$, and $h(x,y)=(x+2,1-y^2).$ I want ...
00GB's user avatar
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Does $h$ exhibits a homeomorphism between $\mathbb R^{J}$ and itself in the Box Topology?

Does the proof here (Does the given $h$ exhibit the homeomorphism between $\mathbb{R}^{\omega}$ and itself with box topologies?) works when we are dealing with $\mathbb R^J$ instead of $\mathbb R^{\...
Janbazif's user avatar
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The range of the function $F$ is $S^2\setminus \{\textbf{n}\}$

Let $S^2:=\{(x,y,z)\in \mathbb{R}^3 \mid x^2+y^2+z^2=1\}$ be the unit sphere, $\textbf{n}:=(0,0,1)$ the northpole of $S^2$ and $\textbf{s}:=(0,0,-1)$ the southpole of $S^2$. Let $F:\mathbb{R}^2\...
Mary Star's user avatar
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Prove $f:[a,b]\to [a,b]$ is a homeomorphism, then $a$ and $b$ are fixed points or $f(a)=b$ & $f(b)=a$

Prove $f:[a,b]\to [a,b]$ is a homeomorphism, then $a$ and $b$ are fixed points or $f(a)=b$ & $f(b)=a$ Hello. I've been struggling with this question. I found something related: Continuous ...
Hackerman's user avatar
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Continuous group homomorphism

Let A be the set of matrix of type \begin{bmatrix} a&0\\ 0 & a^{-1} \end{bmatrix} where a is a positive number. How to show that if $\phi$ is a Group homomorphism from A to $\mathbb{R}...
Tanutanu's user avatar
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Prove we cannot make a homeomorphism from [0, 1] to ℝ by contradiction. [duplicate]

So start off contradiction proofs by assuming the opposite. So we assume f is a homeomorphism from ℝ to [0, 1]. Since f is a surjection, there exists some 𝑎 ∈ ℝ with $f(𝑎) = 0$. Let $x_1 = 𝑎 – 1$ ...
User's user avatar
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Lifting of arbitrary map to a covering space

Let $(\tilde{X},p)$ be a covering space of $X$, $\tilde{x}_{0}\in\tilde{X}$, $x_{0}=p(\tilde{x}_{0})$, $y_{0}\in Y$ and $\varphi :(Y,y_{0})\rightarrow (X,x_{0})$. Under what conditions does there ...
user782709's user avatar
2 votes
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168 views

Continuous manifold dimension

First time encountering manifolds in general. A smooth manifold $M$ is said to be of dimension $n$ if its (complete) atlas is of dimension $n$. Now, what if we replace smooth by continuous ? Does the ...
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Mapping Cylinder Neighbourhoods: Examples

So, yesterday I was trying to do Exercise 2.2.35 from Hatcher's Algebraic Topology, which goes like this: 35. Use the Mayer-Vietoris sequence to show that a nonorientable closed surface, or more ...
StormyTeacup's user avatar
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1 vote
1 answer
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How do we need to apply the fundamental theorem of homomorphisms here?

Let $d\in\mathbb N$, $\alpha\in\mathbb N\uplus\{\infty\}$, $\Omega$ be a bounded $d$-dimensional properly embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary, $\nu_{\partial\Omega}$ denote ...
0xbadf00d's user avatar
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2 answers
71 views

How to show that subspaces of finite-dimensional space are direct sum?

Let $W$ be finite-dimensional vector space over the $\mathbb{R}$ and its subspaces are $U,V$. Let $X^*=\operatorname{Hom}(X,\mathbb{R})$ be a dual space to space $X$ which consists of all linear ...
Novice's user avatar
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1 vote
1 answer
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An isomorphism between two normed vector spaces with the same finite dimension is an homeomorphism

I'm trying a simple proof of this fact: An isomorphism between two normed vector spaces with the same finite dimension is an homeomorphism. I've tried in this way (everything seems to be ok, I ask ...
Nameless's user avatar
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1 answer
1k views

Homeomorphism from a quotient of $D^{2}$ in $\mathbb{R}^{2}$ to $S^{2}$ in $\mathbb{R}^{3}$

Let \begin{align*} D^{2}&=\left\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\leq1\right\}\text{ and } \\ S^{2}&=\left\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}=1\right\} \end{align*} be endowed ...
Representation's user avatar
2 votes
0 answers
391 views

Why can't an embedding have self-intersections?

If I understood it correctly an embedding is an immersion $\phi:M \rightarrow N$ that is a homeomorphism and where $\phi (M) \subset N$. To which condition contradicts an immersion with a self-...
lea5619's user avatar
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1 answer
108 views

removing points, connected component

I have a short question regarding the following: $X = S^1 \times \{ 0,1 \} = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, z \in \{ 0,1 \} \} \subset \mathbb{R}^3$, $B_2 = X /_{\sim}, (1,0,0) \sim (1,...
lea5619's user avatar
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1 answer
602 views

Continuous function whose inverse is not continuous [duplicate]

Suppose $f:\mathbb{R}^n\to\mathbb{R}^n$ is bijective and continuous. Is it possible that $f^{-1}$ is not continuous? I can prove that for $n=1$ it is not possible, i.e. if $f:\mathbb{R} \to \mathbb{R}...
Aditya Ghosh's user avatar
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0 answers
279 views

Topology: Homeomorphism from R^2 to itself preserving standard topology

I need to prove that there exists a homeomorphism f from R2 to itself with its standard topology such that: when considering any pair of three distinct points (x1, x2, x3) and (y1, y2, y3), f maps (...
Lupetto1927's user avatar
2 votes
1 answer
124 views

$\alpha (a + b) = \alpha (a) + \alpha (b)$ for all $a,b \in \mathbb{R}.$ show that $\alpha$ is a linear transformation.

The question is: Let $\alpha: \mathbb{R} \rightarrow \mathbb{R}$ be a cts. function which satisfies $\alpha (a + b) = \alpha (a) + \alpha (b)$ for all $a,b \in \mathbb{R}.$ show that $\alpha$ is a ...
Intuition's user avatar
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3 votes
2 answers
339 views

Example of a continuous affine group action

Let $G$ be any locally compact group and $H$ be a compact group. We know that a map $F: G \rightarrow G$ is called affine if there exists some $\alpha \in G$ and an automorphism $\Lambda:G\...
Carl Butcher's user avatar
2 votes
1 answer
131 views

$*$-homomorphism of $C^*$-algebras and representations

For $i=1,2$, let $\mathcal{A}_i$ be an abstract $C^*$-algebra and $\pi_i : \mathcal{A}_i \rightarrow \mathcal{B}(\mathcal{H}_i)$ a $C^*$-representation. Let $\alpha: \mathcal{A}_1 \rightarrow \mathcal{...
user avatar
3 votes
0 answers
176 views

When is a surjective homomorphism between two unital Banach algebras bounded?

Let $A$ and $B$ be unital Banach algebras and $\theta : A\to B$ a surjective homomorphism between these two spaces. What is a sufficient requirement for $\theta$ being bounded, and how would a proof (...
Markus Klyver's user avatar
1 vote
1 answer
68 views

Proving a condition at which a function is not a homomorphism

Let $A \subset R^k$ be an open set and $f:A \to R^n$ be a continuous injection $(k < n)$. The task is to prove that: $f:A \to f(A)$ is not a homeomorphism if and only if there exists a sequence $...
Gabi G's user avatar
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0 votes
1 answer
65 views

Modulus of continuity under homeomorphisms

Let $f:C:=[0,1]^{d}\longrightarrow \mathbb{R}$ be continuous, and define the (usual) modulus of continuity of $f$ of order $\epsilon$ as $\omega(f;\epsilon):=\sup\{|f(x)-f(y)|:x,y\in C, \|x-y\|\leq \...
user123043's user avatar
2 votes
1 answer
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Let $X$ and $Y$ be connected spaces, then $X\times Y$ is connected

I have 2 questions about this proof of connected topological spaces. Let $X$ and $Y$ be connected spaces, then $X\times Y$ is connected. Proof. Let $p=(x_1,y_1)\in X\times Y,q=(x_2,y_2)\in X\times ...
user avatar
3 votes
1 answer
383 views

A homeomorphism on a dense set in Hausdorff space

Let $X$ be a Hausdorff space, $D \subset X$ be a dense set, and $f:X \rightarrow Y$ be a continuous function such that $f|_D:D \rightarrow f(D)$ is a homeomorphism. Show that $f(X \setminus D) \...
mbrg's user avatar
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0 votes
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Let $\varphi: A \to \mathbb C$ be a non-zero homomorphism. How can we extend it to an homomorphism $\psi: \overline A \to \mathbb C$?

Let $X$ be a Hausdorff compact set. We consider $C(X)$ as a complex Banach algebra with the supremum norm $\| f \|_X = \sup_{x \in X} |f(x)|$. Let $A$ be a Banach function algebra on $X$, i.e $A \...
user 242964's user avatar
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1 vote
1 answer
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Implication about homeomorphism

Let $f:(X,\tau_1) \to (Y,\tau_2) $ be a bijection and $A \subseteq X.$ Then TFAE a)f is a homeomorphism b) $f(\overline A) = \overline {f(A)}$ c) $f(int(A)) = int(f(A))$ Which means we should show ...
usereb's user avatar
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a complete and an incomplete metric space be homeomorphic

I wonder how to find such an example in the title. I already know that since it is a bijection, I have to choose different metrics anyway. but I do not know how to go on. Is it about a metric induced ...
Zihao Li's user avatar
3 votes
2 answers
176 views

Proving that $\sqrt{A}^2=A$

Let $A \in \mathfrak{U}$ where $\mathfrak{U}$ is a $C^*$ algebra and let $A=A^*$ with $\sigma(A)\subseteq \mathbb{R}^+$. Let $f := \sqrt{\cdot}\in \mathcal{C}(\sigma(A))$ where $\mathcal{C}(X)$ is the ...
R.W's user avatar
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0 answers
62 views

Continuous Homomorphism of Matrix Groups - counterexample

Let {${G={\begin{bmatrix} 1 & n \\ 0 & 1 \\ \end{bmatrix}} \in SUT_2(\mathbb R) : n\in\mathbb Z}$}. $$\\$$ For any irrational number r$\in\mathbb R-\mathbb Q$, a function is given with $$\...
Ayneeldor's user avatar
2 votes
2 answers
191 views

Let $f:\mathbb R \longrightarrow \mathbb R$ be continuous which is also an additive homomorphism

Let $f:\mathbb R \longrightarrow \mathbb R$ be continuous which is also an additive homomorphism, that is, $f( x+ y)= f( x )+f(y) $ for all $x,y\in \mathbb R$ then $f( x)= \lambda x$ where $\lambda= ...
Pranita Gupta's user avatar
2 votes
1 answer
166 views

What is the difference between $(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})$ and $ S^{1}\times S^{1}\times S^{1}$.

What is the difference between $(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})$ and $S^{1}\times S^{1}\times S^{1}$. Where their homology groups are: If $\;\;\;\;(S^{2}\times S^{1})\#...
nill's user avatar
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2 votes
1 answer
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Definition of a continuous group homomorphism

I was reading about Lie groups, and whenever they are defining a Lie group homomorphism they state it is a group homomorphism between Lie groups that is continuous. I tried looking for a definition of ...
Cami77's user avatar
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1 answer
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Charakterisation of Homomorphismen from $\mathbb{R} $ to $\mathbb{T}$ [closed]

I wonder if there is a characterisation of all continous Homomorphismen from $(\mathbb{R}_+, \cdot)$ to $(\mathbb{T},\cdot)$. With $\mathbb{R}_+=(0, \infty)$ and $ \mathbb{T} $ the unit Circle in $ \...
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