# Combination Theorem for Continuous Mappings/Topological Ring

Jump to navigation
Jump to search

## 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.