CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
symmetric monoidal (∞,1)-category of spectra
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 are continuous functions.
Similarly a topological field is a topological ring whose underlying ring is in fact a field and a topological algebra is a topological ring under a base topological ring (a topological associative algebra).
Shouldn’t we demand that reciprocation be continuous in a topological field? Or does this somehow follow automatically? -Sridhar Ramesh
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.