Combination Theorem for Continuous Mappings/Topological Ring

Theorem

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

Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.

Let $\lambda \in R$.

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

Then the following results hold:

Sum Rule

$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

Translation Rule

$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

Negation Rule

$-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

Product Rule

$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

Multiple Rule

$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous

$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

Also see

• Combination Theorem for Continuous Mappings to Topological Group
• Combination Theorem for Continuous Mappings to Topological Semigroup