# Topological Space

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.

### Examples

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. |

### Prerequisites

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. |