View source for Point-set topology

== Introduction ==
Point-set topology, also known as general topology, is a branch of [[Mathematics|mathematics]] that studies the properties of topological spaces and the abstract concepts defined on them. It serves as a fundamental tool in many areas of mathematics, including [[Algebraic topology|algebraic topology]], [[Differential geometry|differential geometry]], and [[Functional analysis|functional analysis]].

== Definition of a Topological Space ==
A [[Topological space|topological space]] is a set endowed with a structure, called a topology, which allows the definition of continuous deformation of subsets. This structure is given by a collection of subsets, known as open sets, which satisfy the following conditions:
1. The empty set and the whole space are open.
2. Any union of open sets is open.
3. The intersection of any finite number of open sets is open.

== Basis for a Topology ==
A basis (or base) for a topology on a set X is a collection B of subsets of X such that:
1. For every x in X, there is at least one basis set containing x.
2. If x belongs to the intersection of two basis sets B1 and B2, then there is a basis set B3 containing x that is a subset of the intersection of B1 and B2.

[[Image:Detail-59748.jpg|thumb|center|Photograph of a 3D printed model demonstrating a complex topological structure.|class=only_on_mobile]]
[[Image:Detail-59749.jpg|thumb|center|Photograph of a 3D printed model demonstrating a complex topological structure.|class=only_on_desktop]]

== Continuous Functions ==
In point-set topology, a function between two topological spaces is called [[Continuous function|continuous]] if the inverse image of every open set is open. This definition generalizes the concept of continuity from calculus.

== Homeomorphisms ==
A [[Homeomorphism|homeomorphism]] is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space.

== Separation Axioms ==
Separation axioms are conditions in point-set topology that a topological space may satisfy. The most common are:
1. T0 space: For any distinct points, there is an open set containing one but not the other.
2. T1 space: For any distinct points, there are open sets containing each point but not the other.
3. T2 space, or Hausdorff space: For any distinct points, there are disjoint open sets containing each point.

== Compactness ==
A topological space is said to be [[Compact space|compact]] if each of its open covers has a finite subcover. Compactness is a topological property that generalizes the notion of a set being closed and bounded in Euclidean space.

== Connectedness ==
A topological space is [[Connected space|connected]] if it cannot be divided into two disjoint nonempty open sets. This concept is related to the intuitive idea of a space that cannot be broken into pieces.

== Metric Spaces ==
A [[Metric space|metric space]] is a set where a notion of distance (called a metric) is defined. Metric spaces are an important class of topological spaces, and all metric spaces are topological spaces.

== See Also ==
* [[Algebraic topology]]
* [[Differential topology]]
* [[Functional analysis]]
* [[Metric space]]
* [[Topological manifold]]

[[Category:Mathematics]]
[[Category:Topology]]
[[Category:Point-set topology]]