Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule
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Theorem
Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.
Let $g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ be a continuous mappings.
Let $-g : S \to R$ be the mapping defined by:
- $\forall x \in S: \map {\paren {-g} } x = -\map g x$
Then
- $-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Proof
By definition of a topological ring:
- $\struct {R, +, \tau_{_R} }$ is a topological group.
From Inverse Rule for Continuous Mappings to Topological Group:
- $-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
$\blacksquare$