For example, self-adjoint elements of a commutative C*-algebra correspond to real-valued continuous functions.
Also, projections (i.e. self-adjoint idempotents) correspond to indicator functions of clopen sets.
For example, the coproduct of spaces is the disjoint union and thus corresponds to the direct sum of algebras, which is the product of C*-algebras.
In a more specialized setting, compactifications of topologies correspond to unitizations of algebras.
There are certain examples of properties where multiple generalizations are possible and it is not clear which is preferable.
A further development in this is a bivariant version of K-theory called KK-theory, which has a composition product
of which the ring structure in ordinary K-theory is a special case.