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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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 ...
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1answer
38 views

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 ...
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2answers
34 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 ...
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37 views

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 ...
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159 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 ...
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60 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-...
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36 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,...
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93 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}...
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92 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 (...
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77 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 ...
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36 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\...
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1answer
63 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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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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39 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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43 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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87 views

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 ...
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205 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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24 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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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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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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137 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 ...
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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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119 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= ...
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1answer
100 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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1k 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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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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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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1answer
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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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296 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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899 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
111 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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257 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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48 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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1answer
345 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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151 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
102 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
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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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327 views

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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3answers
1k 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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3answers
73 views

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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31 views

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
67 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
93 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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1answer
335 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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1answer
124 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
359 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) ...