Say I have some topology $\mathcal{T}$ on $X = \lbrace a,b,c \rbrace$ and a disconnected subset $A \subset X$. Can it be true that any arbitrary pair of sets - say $U$ and $V$ - that is a separation of $A \in X$ will satisfy $U \cap V \cap (X - A) \neq \emptyset$ ? I have the definition of separa...

A group $G$ is called co-Hopfian if it is not isomorphic to a proper subgroup of itself; equivalently, every injective group homomorphism $\varphi : G \to G$ is surjective and hence an isomorphism. Examples of co-Hopfian groups include finite groups, $\mathbb{Q}$, $\mathbb{Q}/\mathbb{Z}$, and fun...

Let $G$ be a group. We say that $G$ is a Hopfian(Co-Hopfian) group, if every epimorphism(monomorphism) of $G$ if a monomorphism(epimorphism). Can someone give me some applications of Hopfian and co-Hopfian groups in geometry...?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In what kind of group could I search for a counterexample ?

trying to think of any residually finite group which is not Hopf. Any help?

Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context in which Hopf first used this concept, and a reference for this?