Criterion for Ring with Unity to be Topological Ring

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Theorem

Let $\struct {R, +, \circ}$ be a ring with unity.

Let $\tau$ be a topology over $R$.

Suppose that $+$ and $\circ$ are $\tau$-continuous mappings.

Then $\struct {R, +, \circ, \tau}$ is a topological ring.

Proof

As we presume $\circ$ to be continuous, we need only prove that $\struct {R, +, \tau}$ is a topological group.

As we presume $+$ to be continuous, we need only show that negation is continuous.

As $\struct {R, \circ}$ is a semigroup and $\circ$ is continuous:

$\struct{R, \circ, \tau}$ is a topological semigroup.

From Identity Mapping is Homeomorphism, the identity mapping $I_R : \struct {R, \tau} \to \struct {R, \tau}$ is continuous.

From Multiple Rule for Continuous Mappings to Topological Semigroup, the mapping $\paren{- 1_R} \circ I_R : R \to R$ defined by:

$\forall b \in R : \map {\paren {\paren {-1_R} \circ I_R} } b = \paren {-1_R} \circ b$

is continuous.

From Product with Ring Negative, for each $b \in R : -b = \paren {-1_R} \circ b$.

Hence negation is continuous.

$\blacksquare$