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A topological ring is a ring internal to Top, a ring object in Top:
a topological space $R$ equipped with the structure of a ring on its underlying set, such that addition $+$ and multiplication $\cdot$ are continuous functions.
Remarks:
The structure of a topological ring $(R,+,\cdot)$ makes $R$ a uniform space.
It is automatic that negation in a topological unital ring is continuous, since it is the operation of multiplication with $-1$, and multiplication is continuous in each argument. Hence for $(R,+,\cdot)$ a topological ring, then $(R,+)$ is a topological group.
A topological field is a topological ring $K$ whose underlying ring is in fact a field and such that reciprocation $(-)^{-1}: K \setminus \{0\} \to K \setminus \{0\}$ is continuous. This latter condition is the same as demanding that the subspace topology on $K \setminus \{0\}$ induced by the embedding $K \setminus \{0\} \hookrightarrow K$ coincide with the subspace topology induced by the embedding $K \setminus \{0\} \to K \times K: x \mapsto (x, x^{-1})$. More at topological field.
In a topological ring, the closure of $\{0\}$ is an ideal. It follows that for a topological field $F$, either $0$ is a closed point (so that $F$ is $T_1$ and therefore completely regular Hausdorff, by standard arguments in the theory of uniform spaces), or is a codiscrete space.
A topological algebra over a topological ring $R$ is a topological ring $S$ together with a topological ring map $R \to S$ that makes $S$ an $R$-algebra at the underlying set level (a topological associative algebra).
The real numbers form a topological field.
Any pseudocompact ring such as the completed group ring of a profinite group is a topological ring.
For any prime $p$, the ring of p-adic integers is a topological ring.
A Banach algebra is in particular a topological algebra, hence a topological ring. Hence so is a C-star-algebra.
Lecture notes include
See also
Wikipedia, Topological ring
Last revised on May 30, 2017 at 05:42:30. See the history of this page for a list of all contributions to it.