Topology

Topology is a branch of mathematics concerned with the properties of space that are preserved under continuous transformations. These properties include concepts such as continuity, compactness, and connectedness. Topology is often referred to as "rubber-sheet geometry" because it involves the study of properties that remain unchanged when objects are stretched or deformed, but not torn or glued.

Basic Concepts

Topological Spaces

A topological space is a set equipped with a topology, a collection of open sets that satisfy certain axioms. These axioms ensure that the union of open sets is open, the intersection of a finite number of open sets is open, and the set itself and the empty set are open.

Continuous Functions

A function between two topological spaces is continuous if the preimage of every open set is open. This generalizes the concept of continuity from calculus.

Homeomorphisms

A homeomorphism is a continuous function with a continuous inverse. Two spaces that are homeomorphic are considered topologically equivalent.

Compactness and Connectedness

Compactness is a property that generalizes the notion of a set being closed and bounded. A space is compact if every open cover has a finite subcover. Connectedness refers to a space that cannot be divided into two disjoint nonempty open sets.

Examples of Topological Objects

Mobius Strip

The Mobius strip is a surface with only one side and one boundary component. It is a classic example of a non-orientable surface.

Torus

A torus is a surface shaped like a doughnut. It is a common example of a compact, orientable surface.

Knot Theory

Knot theory is a branch of topology that studies mathematical knots. A knot is an embedding of a circle in 3-dimensional space.

Applications

Topology has applications in many areas of mathematics and science, including differential geometry, algebraic topology, and dynamical systems. It is also used in computer science for data analysis and in physics for studying the properties of space-time.

Gallery

• 

  A Möbius strip

• 

  Map of the Seven Bridges of Königsberg

• 

  Morphing between a mug and a torus

• 

  Spot the cow in a topological puzzle

• 

  Topology joke

Related Pages

• Algebraic topology
• Differential topology
• Knot theory
• Graph theory