# 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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### 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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