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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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\...
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42 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{...
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The induced homomorphism $f_{*} : \pi (X, p) \rightarrow \pi(Y, fp)$ where $f: X \rightarrow Y $ is a continuous mapping.

Consider the induced homomorphism $f_{*} : \pi (X, p) \rightarrow \pi(Y, fp)$ where $f: X \rightarrow Y $ is a continuous mapping. Are the following statements true or false? $(a)$ If $f$ is onto, ...
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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 (...
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Unique homomorphic extension, Free generation : Natural Numbers Vs Integers

Unique homomorphic extension says that for a freely generated set $X_+$(say generated from set $X$ and set of functions $F$ ), given a map $h: X \rightarrow B$ (where B is a set with a set of ...
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1answer
20 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 $...
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38 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 \...
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52 views

What's the homeomorphism that should be used in this proof about connected spaces?

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 ...
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1answer
144 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) \...
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19 views

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 \...
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1answer
33 views

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 ...
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86 views

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 ...
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2answers
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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 ...
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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 $$\...
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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= ...
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1answer
71 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})\#...
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1answer
828 views

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 ...
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1answer
17 views

Charakterisation of Homomorphismen from $\mathbb{R} $ to $\mathbb{T}$

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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Example of a continuous function from $[0,1]$ to a Hausdorff space $X$ that is not one-one but the image is homeomorphic to $[0,1]$.

Suppose that $X$ is a Hausdorff space and $\alpha : [0, 1] \to X$ is a continuous function. If $\alpha$ is one-one, then prove that the image of $\alpha$ is homeomorphicto $[0, 1]$. Give an example ...
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Same homorphism group fundamental [closed]

Prove that two continuous mappings φ, ψ:X→Y, with φ(x0)=ψ(x0) for some point x0∈X, induce the same homomorphism from π(X,x0) to π(Y,φ(x0)) if φ and ψ are homotopic relative to x0.
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Can I define a linear map on the set $\{X\in \mathcal{S}^{n\times n}: X\succeq 0, \operatorname{X}=1 \}$?

The definition of a linear map $\phi$: $\mathcal{V} \mapsto\mathcal{W}$ is: $\forall u, v \in \mathcal{V}$ and $\forall a,b \in \mathbb{F}$ such that $\phi(au+bv) = a\phi(u) + b\phi(v)$. Can ...
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252 views

Distance between zeros of continuous function

Dears, Let $f(x):=1+\sum_{k=1}^{n}a_{i}b_{i}^{k}$, for each $x\in[a,b]$, where $a_{i}$ are real numbers (not nulls) and $b_{i}>0$. Assume that $f$ has, at least, two ceros in $[a,b]$. Then, I want ...
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451 views

homeomorphism from interval $[a,b]$ to $[0,1]\subset \mathbb{R}$

I need to show that every interval $[a,b]$ is homeomorph to $[0,1]\subset \mathbb{R}$. I've found this answer but it only deals with open sets, and I need an answer that deals with closed sets.
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1answer
94 views

Does this line preserving, continuous bijection on an equilateral triangle exist?

I'm trying to define a continuous bijection on the points (x/y coordinates) of an equilateral triangle. The vertices and the midpoints of each edge need to remain fixed. The (6) points on the edges ...
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214 views

Symbol for continuous dual

Let $X$ be a vector space. The dual is the set of all linear forms from $X$ to $\mathbb{F}$.(can be $\mathbb{R}$ or $\mathbb{C}$) That is, $X^* = \operatorname{Hom}(X,\mathbb{F})$. If $X$ is also ...
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36 views

Prove why a closed, continuous bijective map is a homeomorphism.

Prove why a closed, continuous bijective map is a homeomorphism. I'm trying to see if $f^{-1}$ is continuous but nothing happens.
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183 views

to find basis of homomorphism

Compute $Hom(V,W)$ and also determine its dimension over $F$ where $V$ and $W$ are vector spaces over the Field $F$ given that $V=\mathbb R^2, W=\mathbb R^2, F=\mathbb R$ I have done this: $V=\{v1=(1,...
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Existence of a linear structure in a space of homomorphisms

Let $H$ be an (uncountable) set of homomorphisms from an additive group $\Gamma$ to $\mathbb{C}$. Is it possible to define a linear structure on $H$ (over $\mathbb{R}$ or $\mathbb{C}$)? In case it is ...
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1answer
82 views

topological isomorphism between a group and product of its subgroups

I have stumbled upon the following question: Let $G$ be a $\sigma$-compact, locally compact Hausdorff group with $N$ and $H$ closed normal subgroups of $G$. Also $$N\cap H= \{e\}$$ and $$G=NH .$$ Then ...
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1answer
92 views

LCA - groups under continuous homomorphisms

can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image Let $A$ and $B$ be LCA-groups and $H$ a (not ...
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Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show that ...
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1answer
149 views

Proof that $\mathbb Q_p$ is unique up to unique isomorphism preserving the absolute values

On pages 58-59 of Gouvea's $p-$adic Numbers: An Introduction, he gives the following proof that the field $\mathbb Q_p$, constructed using equivalence classes of Cauchy sequences, is unique up to ...
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Does the ring of continuous functions determine $\mathbb R^n$?

I have two related questions which are just making the question asked in the title more specific: (a) Is every ring homomorphism (or maybe $\mathbb R$-algebra homorphism) between rings of the form $\...
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728 views

Continuous homorphisms between topological groups.

Let $G,H$ be topological groups and let $\rho: G \rightarrow H$ be a homomorphism of topological groups. I understand this to mean that $\rho$ preserves group structure and is a map between the ...
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What are the subgroups of $(S^1)^n$ isomorphic to the standard copy of $(S^1)^k$?

Let $H$ be the set of all subgroups of the $n$-dimensional torus $(S^1)^n$ that are isomorphic by an element of $\operatorname{Aut}((S^1)^n)$, the set of continuous automorphisms of $(S^1)^n$, to the ...
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What is $\operatorname{Hom}((S^1)^k , (S^1)^n)$?

I am trying to find $\operatorname{Hom}_{\rm gp}((S^1)^k , (S^1)^n)$ , which is the set of continuous group homomorphisms from the $k$ dimensional torus to the $n$ dimensional torus where $1 \leqslant ...
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Prove that the “additive” operation of the module ($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=(\mathbb{Z}_{p}^{*},\ \mathbb{Z}_{p-1},\cdot)$ in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar "...
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1answer
51 views

compact inverse is compact in canonical homomorphism

Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there ...
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1answer
58 views

Let $f:(\mathbb F,+)\to (\mathbb F,+)$ be a non-zero homomorphism. Then which is true?

Its (may be) asking about the generalization of this question. Let $f : (\Bbb F,+) \to (\Bbb F,+)$ be a non-zero homomorphism. Pick out the true statements: a. $f$ is always one-one. b. $f$ is ...
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221 views

Proving linearity implies (or can imply under opportune conditions) lower semicontinuity

A function $f:X\to\mathbb{R}$, with $X$ being a topological space, is termed as lower semicontinuous (lsc) at $x_0\in X$ if: $$\forall\epsilon>0\,\,\exists V\text{ an open neighborhood of }x_0:x\in ...
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91 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
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1answer
206 views

Is there a natural (non-trivial) topology for the automorphism group of a locally compact abelian group?

I've been thinking about automorphism groups of locally compact abelian groups somewhat over the last few days. It's not hard to see that in the case of the torus and integers, the (continuous) ...
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2answers
208 views

What are the continuous automorphisms of $\Bbb T$?

I wanted to check my reasoning on this problem. From standard Pontrjagin duality arguments, it's not hard to see that the continuous homomorphisms of the torus (to itself) are nothing more than the ...
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Is the tensor product of two representations a representation?

I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie ...
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740 views

Induced Lie algebra homomorphism from Lie group homomorphism: left translation

A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$. Which Lie algebra homomorphism ...
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218 views

Continuous, differentiablee and continuous isomorphism, homomorphism

I have very big confusions on Continuous, differentiable and continuous isomorphism, homomorphism of a FUNCTION. I am seriously looking a single example, which can explain the following clearly. ...
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634 views

Topology: homeomorphism with restricted domain

In topology, if I restrict the domain of a homeomorphism and find who's the image of the restricted homeomorphism, the domain and the image are still homeomorphs? For example... Prove that if $X$ and ...
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29 views

Homomorphisms and conjugacy

(a) Let $\theta:G→H$ be a homomorphism and let $x,y \in G$. Prove that if $x$ and $y$ are conjugate in $G$ then $\theta(x)$ and $\theta(y)$ are conjugate in $H$. (b) By considering the homomorphism $\...
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209 views

Homomorphism and kernel proof help

Consider the map $\phi: \mathbb R^* \to \mathbb R^*$ (under multiplication) defined by $\phi(x) = |x|$. Prove that $\phi$ is a homomorphism and find $\ker \phi$. Proof: Now $\phi(xy) = |xy| = |x||y| =...
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1answer
67 views

Homotopy between two homomorphisms and homology

If I have two chain complexes $C$ and $D$ and I suppose that there is a homotopy between $\phi, \psi:C \rightarrow D$ (i.e there is a sequence of homomorphisms $(K_n: C_n\rightarrow D_{n+1})$ such ...