A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing
with topological spaces, a disconnectivity
is interpreted as a hole in the space. Examples of holes are things like the "donut
hole" in the center of the torus, a domain removed
from a plane, and the portion missing from Euclidean
space after cutting a knot out from it.

There are many ways to measure holes in a space. Some holes are picked up by homotopy groups that are not detected by homology groups,
and some holes are detected by homology groups
that are not picked up by homotopy groups. (For
example, in the torus, homotopy
groups "miss" the two-dimensional hole that is given by the torus
itself, but the second homology group picks that
hole up.) In addition, homology groups don't detect
the varying hole structures of the complement of knots in
three-space, but the first homotopy group (the
fundamental group) does.