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:

Also see

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