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**Topology** is (roughly) the study of properties invariant under "continuous transformation".

More formally:

Let *S*, *T* be sets with a structure for which the idea of continuity makes sense. e.g. Subsets of **R**^{n}, metric spaces, ...

Then a bijection *f* from *S* to *T* is called a **homeomorphism** or **topological isomorphism** if both *f* and *f*^{-1} are continuous. We then write *S* *T*.

In Klein's formulation, the set of all such maps from a space to itself is a group and topology is the associated geometry.

**Examples**

- A line and a curve are homeomorphic:
- A
**circle***S*^{1}and a**knot***K*(subsets of**R**^{3}) are homeomorphic -- even though one cannot be deformed into the other (in**R**^{3}).

Strangely,**R**^{3}-*S*^{1}and**R**^{3}-*K*are not homeomorphic. - A closed (including its boundary) disc and closed unit square are homeomorphic.
- A
**sphere**(the surface) and the**surface of a cube**are homeomorphic.

**"Proof"**

Put the cube at the centre of the sphere and project from the centre. - A
**sphere**and a**torus**are not homeomorphic.

**"Proof"**

Removing a circle from a sphere always splits it into two parts -- not so for the torus. - A cylinder S1 × I and the space made by gluing a strip after a full turn are homeomorphic -- even though (cf example 2) one cannot be deformed into the other.
- The
**Möbius band**is a "space" made by gluing together the ends of a strip after a half turn. - The
**Real projective plane****R***P*^{1}= {the set of lines through**0**in**R**^{3}}

If we take an "upper hemisphere", this meets each line in a unique point which we may take as the representative of that point of the projective plane -- except for opposite points on the equator which have to be "identified".

Hence the Real Projective Plane can be regarded as a disc which opposite points on its boundary identified.

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