Topological Space

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A topological space is a set with a collection of subsets T that together satisfy a certain set of axioms defining the topology of that set.

Topological space is a college-level concept that would be first encountered in a topology course covering point-set topology.


Euclidean Space: Euclidean space of dimension n is the space of all n-tuples of real numbers which generalizes the two-dimensional plane and three-dimensional space.
Möbius Strip: A Moebius strip is one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving it half a twist, and then reattaching the two ends.
Projective Plane: The projective plane is the set of lines in the Euclidean plane that pass through the origin. It can also be viewed as the Euclidean plane together with a line at infinity.
Sphere: A sphere is the set of all points in three-dimensional space that are located at a fixed distance from a given point.
Torus: A torus is a closed surface containing a single hole that is shaped like a doughnut.


Set: In mathematics, a set is a finite or infinite collection of objects in which order has no significance and multiplicity is generally also ignored.
Topology: (1) As a branch of mathematics, topology is the mathematical study of object's properties that are preserved through deformations, twistings, and stretchings. (2) As a set, a topology is a set along with a collection of subsets that satisfy several defining properties.

Classroom Articles on Point-Set Topology

  • Closed Set
  • Open Set
  • Homeomorphism
  • Point-Set Topology
  • Neighborhood
  • Subspace

  • Classroom Articles on Topology (Up to College Level)

  • Dimension
  • Metric Space
  • Metric
  • Projective Space