Combination Theorem for Continuous Mappings/Topological Division Ring

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Theorem

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

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


Let $\lambda \in R$.

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


Let $U = S \setminus \set{x : \map g x = 0}$

Let $g^{-1} : U \to R$ denote the mapping defined by:

$\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$

Let $\tau_{_U}$ be the subspace topology on $U$.


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.


Inverse Rule

$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous.


Also see