# 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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### 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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