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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.
Here are some examples of typical questions in topology: How many holes are there in an object? How can you define the holes in a torus or sphere? What is the boundary of an object? Is a space connected? Does every continuous function from the space to itself have a fixed point?
Topology is a relatively new branch of mathematics; most of the research in topology has been done since 1900. The following are some of the subfields of topology.
General Topology or Point Set Topology: General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense. Combinatorial Topology. Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology, and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic. This is the number (V - E + F), where V, E, and F are the number of vertices, edges, and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any “triangulation” of a sphere will have an Euler characteristic of 2.
Algebraic Topology: Algebraic topology also considers the global properties of spaces, and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups.
Topological space as explained are charaterized in terms of holes or handles. A sphere has zero hole. Toplogy has given birth to a new science to study the shape of the universe. We are three dimensional being who inhabit a four dimensional world ( space-time). We can not leave our universe to see what is the shpae or structure of it. So topological features can tell how our universe looks entirely within itself.
All topological spaces which have indentical number of holes are of same kind. They can be transformed onto each other by continuous mapping or functions.