----- > [!proposition] Proposition. ([[left and right translations preserve connectedness and compactness for subspaces of topological groups]]) > Let $G$ be a [[topological group]] with $g \in G$, and let $C \subset G$ a [[connected]] [[subspace topology|subspace]] of $G$. Then the sets $gC:= \{ gc : c \in C \} \text{ and } Cg := \{ cg : c \in C \}$ > are [[connected]]. > > Same deal for [[compact|compactness]], and indeed any topological invariant. > [!basicexample] > Taking $G=(\mathbb{R}, +)$, we have that $4+(1,2)=\{ 4+x : x \in (1,2) \}=(5,6)$ is [[connected]]. > [!proof]- Proof. ([[left and right translations preserve connectedness and compactness for subspaces of topological groups]]) > We implicitly use [[product and subspace topologies commute]] and [[restriction of continuous function is continuous]]. > > Since the group operation $\begin{align} \cdot: & G \times G \to G \\ & (g,h) \mapsto g \cdot h \end{align}$ is [[continuous]], it maps [[connected]] sets to [[connected]] sets ([[continuity preserves connectedness]]). In particular, since $\{ g \} \times C$ is [[connected]] in the [[product topology]] [[a finite cartesian product of connected spaces is connected|as a product of connected subsets]], we have that $\cdot (\{ g \} \times C)=\{ \cdot(g, c): c \in C \}=\{ g \cdot c : c \in C\}=g C$ is connected. Likewise, since $C \times \{ g \}$ is [[connected]], $\cdot (C \times \{ g \})=\{ \cdot(c, g): c \in C \}=\{ c \cdot g : c \in C\}=C g$ is [[connected]]. ^02ccb3 ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```