Subsets of Topological Hausdorff Groups Let $A, B$ be two subsets of a topological Haudorff Group. Need to show that


*

*$\forall B: A$ open $\Rightarrow$ $AB$ open

*$A,B $ compact $\Rightarrow$ $AB$ compact

*$A$ closed and $B$ compact $\Rightarrow AB$ closed

*$A,B $ closed than $AB$ does not necesarry need to be closed 


So far i know that every $T1$ Group is a Hausdorff (right?). I dont know how to do this since I have not subgroups here but subsets...
Anyone an idea?
Thanks

Let $G$ be your group.


*

*$AB=\bigcup_{b\in B}Ab$, which is an union of open sets.

*$AB$ is the image of $\pi\colon A\times B\longrightarrow G$ defined by $\pi(x,y)=xy$ and $A\times B$ is compact.

*There is an answer here.

*Take, in $(\mathbb{R},+)$ (endowed with its usual topology) $A=\mathbb{N}$ and $B=\left\{-n+\frac1n\,\middle|\,n\in\mathbb{N}\right\}$. Then $A$ and $B$ are closed, but $A+B$ is not (it contains the sequence $\left(\frac1n\right)_{n\in\mathbb N}$, but not its limit).

