Homeomorphism: the isomorphisms of topological spaces.



When studying topological spaces, you will come across homeomorphisms, in one of my papers, i presented a paper on "homeomorphism of C-star algbera and their representations" where we defined homeomorphism on a C-star algebra since homeomorphisms deals with continuous functions in topological spaces and cstar algbera can be studied over a case of complex algebra with continuous linear operator on a complex Hilbert space, their relationship of continuity enabled a definite construction of homeomorphism over C-star algbera.
But today we are going to focus on homeomorphisms in regards to topological spaces.
Homeomorphism can simply be defined as isomorphisms in the category of topological spaces i.e they are mappings that preserve all topological properties of a given space. Homeomorphism are also referred to as topological isomorphisms or bicontinuous functions.
When two spaces are binded by homeomorphism we say they are homeomorphic and therefore the same. 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. More reason why the continuous deformation between a coffee mug and a doughnut is regarded as a homeomorphic relationship. Continuous deformation are not necessarily the basis for homeomorphism. Some continuous deformations are not homeomorphism while others are for example the deformation of a line into a point is not homeomorphic. 

Mathematical definition of homeomorphism. 
A function $f:X\rightarrow{Y}$ between two topological spaces is homeomorphism if the following properties is satisfied:
  1. $f$ is a bijection (one-to-one and onto) 
  2. $f$ is continuous 
  3.  the inverse function $f^{-1}$ is also continuous.
We must note that in homeomorphism, the homeomorphic spaces $X$ and $Y$ share the same topological properties. For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well. Also homeomorphism maps open sets to open sets and closed set to closed set. But in the case of metric spaces there are metric spaces that are homeomorphic even though one of them is complete and the other is not. 
Examples of homeomorphism includes:
  1.  The open interval $(a,b)$ is homeomorphic to the real numbers $\mathbb{R}$ for any $a<b$. 
  2. The unit $2$-disk $D^2$ and the unit square in $R^2$ are homeomorphic, since the unit disk can be deformed into the unit square. 
  3.  The chart of a manifold is a homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. 
  4. The graph of a differentiable function is homeomorphic to the domain of the function.
  5.  If $G$ is a topological group, its inversion map $x\mapsto{x^{-1}}$ is a homeomorphism.