Topological divisor of zero

In mathematics, an element of a Banach algebra is called a topological divisor of zero if there exists a sequence of elements of such that

The sequence converges to the zero element, but
The sequence does not converge to the zero element.

If such a sequence exists, then one may assume that for all.
If is not commutative, then is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly.

Examples

If has a unit element, then the invertible elements of form an open subset of, while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero.
In particular, any quasinilpotent element is a topological divisor of zero.
An operator on a Banach space, which is injective, not surjective, but whose image is dense in, is a left topological divisor of zero.

Generalization

The notion of a topological divisor of zero may be generalized to any topological algebra. If the algebra in question is not first-countable, one must substitute nets for the sequences used in the definition.