Combination Theorem for Continuous Mappings/Topological Group

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {S, \tau_{_S} }$ be a topological space.

Let $\struct {G, *, \tau_{_G} }$ be a topological group.


Let $\lambda \in G$.

Let $f, g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ be continuous mappings.


Then the following results hold:


Product Rule

$f * g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.


Multiple Rule

$\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping
$f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.


Inverse Rule

$g^{-1}: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping.


Also see