Prove that homeomorphism is an equivalence relation in the family of topological spaces

For homeomorphisms in graph theory, see Homeomorphism [graph theory].

Not to be confused with homomorphism.

"Topological equivalence" redirects here. For topological equivalence in dynamical systems, see Topological conjugacy.

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος [homoios] = similar or same and μορφή [morphē] = shape or form, introduced to mathematics by Henri Poincaré in 1895.[1][2]

Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.

An often-repeated mathematical joke is that topologists cannot tell the difference between a coffee cup and a donut,[3] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in the cup's handle.

A function f : X → Y {\displaystyle f:X\to Y}   between two topological spaces is a homeomorphism if it has the following properties:

• f {\displaystyle f}   is a bijection [one-to-one and onto],
• f {\displaystyle f}   is continuous,
• the inverse function f − 1 {\displaystyle f^{-1}}   is continuous [ f {\displaystyle f}   is an open mapping].

A homeomorphism is sometimes called a bicontinuous function. If such a function exists, X {\displaystyle X}   and Y {\displaystyle Y}   are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. "Being homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes.

• The open interval [ a , b ] {\textstyle [a,b]}   is homeomorphic to the real numbers R {\textstyle \mathbf {R} }   for any a < b {\textstyle a