2
votes
5answers
117 views

Open subgroups of $\mathbb{R}$ [duplicate]

Let $G$ be a nonempty open subset of $\mathbb{R}$ (with usual topology on $\mathbb{R}$) such that $x,y\in G$ implies that $x-y\in G$. Show that $G=\mathbb{R}$. Clearly $0\in G$. Now how to show that ...
7
votes
0answers
105 views

An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
0
votes
3answers
82 views

group of homeomorphisms subgroup

(a) Let X be a topological space. Prove that the set $Homeo(X)$ of homeomorphisms $f:X \to X$ becomes a group when endowed with the binary operation $f \circ g$ . (b) Let $G$ be a subgroup of ...
2
votes
1answer
361 views

Topological Group, symmetric neighborhood, Hausdorff, disjoint open sets

Let $G$ be a topological group with identity $e$. If $A, B$ are subsets of $G$, we let $A * B$ denote the collection of elements $a * b$ for $a \in A, b \in B$, and we let $A^{-1}$ denote the set of ...
2
votes
1answer
129 views

In topological groups. Is every neighborhood of $e$ supset of a square of a symmetric neighborhood of $e$?

Let $G$ be a topological group, $U$ is a neighborhood of $e$ which is the unit element of $G$. My question is does there exist a neighborhood $H \subseteq U$ of $e$ s.t. $H^{-1}=H$ $H\cdot ...
1
vote
1answer
127 views

topological groups basic facts

How to show that the usual metric with the usual addition is a topological group? Can anybody please explain me briefly about topological groups and the way that I need to approach to this question?
2
votes
2answers
275 views

Locally compact topological group is Normal

How can I prove directly that a locally compact topological group G is normal? I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove ...