Topology

Wikipedia article on topology  says:

• "Topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing."
• "Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation"

Spend some solid amount of time to get a clear picture on Topology. 

Topology is a rich field and has many sub-fields. 

• General topology, also called point-set topology, establishes the foundational aspects of topology and investigates properties of topological spaces and concepts inherent to topological spaces. It defines the basic notions used in all other branches of topology (including concepts like compactness and connectedness).
• Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
• Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
• Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.

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