# 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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### Injective continuous endomorphisms of the unit circle are only the involutions

As done here An injective continuous self-map of the unit circle is a homeomorphism, one can show that any continuous injective homomorphism from the unit circle to itself must be surjective. Now, ...
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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 ...
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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 $y \in \mathbb{R}_{+}$ ?
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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 ...
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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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### 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 $S^{1}\times S^{1}\times S^{1}$.

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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