Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$ ? Also ; does every connected metric space $X$ contains a connected subset which is homeomorphic with $X$ ? UPDATE : So as noticed by @orangeskid ; th...

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is not requested $f(x,0)=x$. My question is this: If $f_t(x):=f(x,t)$ is a homeomorphism for each t...

These questions seem simple, and but I have not found the answer on the web (I have no mathematician in my neighborhood). Does a continuous injective function from $E$ to $E$ have to be surjective, where $E$ is either the $n$-sphere, the $n$-torus, or more generally a compact manifold without bo...

I propose to merge the morphism and homomorphism tags.

Resolved I don't care too much for homomorphism. It lumps together questions about homomorphisms in different algebraic structures, which work quite differently, in general. Therefore, few users will find it convenient to ignore or favorite. The tag for each of these algebraic structures n...