# 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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### 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))}(\...
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### 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 ...
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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 $x_1 = 𝑎 – 1$ ...
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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 ...
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### 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 ...
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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 ...
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### 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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### 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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### 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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### 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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### 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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### $\alpha (a + b) = \alpha (a) + \alpha (b)$ 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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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 $\Lambda:G\... • 370 1 vote 1 answer 92 views ###$*$-homomorphism of$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{... 128 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 (...
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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 $\;\;\;\;(S^{2}\times S^{1})\#... • 179 2 votes 1 answer 2k views ### Definition of a continuous group homomorphism I was reading about Lie groups, and whenever they are defining a Lie group homomorphism they state it is a group homomorphism between Lie groups that is continuous. I tried looking for a definition of ... • 351 0 votes 1 answer 21 views ### Charakterisation of Homomorphismen from$\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} $the unit Circle in$ \... 184 views

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