Mathcenter Forum - Topology

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Topology is the study of shapes and spaces.
What happens if one allows geometric objects to be stretched or squeezed
but not broken?
In fact there’s quite a bit of structure in what remains,
which is the principal subject of study in topology.

Topology studies properties of spaces that are invariant under any continuous deformation.

It is sometimes called "rubber-sheet geometry" because the objects can be stretched
and contracted like rubber, but cannot be broken.

For example, a square can be deformed into a circle without breaking it,
but a figure 8 cannot. Hence a square is topologically equivalent to a circle,
but different from a figure 8.

https://uwaterloo.ca/pure-mathematic...at-is-topology

Topology is used in many branches of mathematics,
such as differentiable equations, dynamical systems, knot theory, and
Riemann surfaces in complex analysis.

It is also used in string theory in physics, and
for describing the space-time structure of universe.

Topology is the mathematical study of the properties that are preserved
through deformations, twistings, and stretchings of objects.

Tearing, however, is not allowed.

A circle is topologically equivalent to an ellipse
(into which it can be deformed by stretching)
and a sphere is equivalent to an ellipsoid.

Similarly, the set of all possible positions of the hour hand of a clock is
topologically equivalent to a circle
(i.e., a one-dimensional closed curve with no intersections that
can be embedded in two-dimensional space),

the set of all possible positions of the hour and minute hands taken together is
topologically equivalent to the surface of a torus
(i.e., a two-dimensional a surface that can be embedded
in three-dimensional space),

and the set of all possible positions of the hour, minute,
and second hands taken together are topologically equivalent to
a three-dimensional object.

https://mathworld.wolfram.com/Topology.html