Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule

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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:

$\forall x \in S: \map {\paren {-g} } x = -\map g x$


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


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.


Also see