# Topological space

A topological space is the basic subject of the sub-discipline of mathematics topology. By introducing a topological structure on a lot to intuitive positional relationships such as " near " and "striving against " from the space of intuition can be transferred to many and very general structures and provided with precise meaning.

## Definition

A topology is a collection of sets T of a basic set X for which the following axioms are composed of subsets (called " open sets " ) are satisfied:

- The empty set and the universal set X are open sets.
- The intersection of a finite number of open sets is an open set.
- The union of any number of open sets is an open set.

A quantity X, together with a topology T to X is a topological space (X, T).

## Basic concepts

### Speech: elements are points, the amount is a space

From the space of intuition, has the term " item" for the elements of the basic amount and the term " ( topological ) space" for the amount that bears the topological structure enforced. Formal correctly is a topological space but the couple from the structure-bearing amount and the structure defining system (the " topology " ) of subsets.

### Dual: completed

A subset of a topological space X, its complement is an open set means complete. If you dualizes formulated above definition and the word " complete", " open " is replaced by ( and cutting and union reversed), an equivalent definition of the term " topological space " through its system of closed sets yields.

### Environments

In a topological space of each point x has a filter U (x) of environments. This allows the intuitive notion of "proximity" grasp mathematically. This concept also can be used as the basis of a definition of topological space.

### Comparison of topologies: coarse and fine

On a fixed set X, one can compare certain topologies T and S together: It's called a topology T " fine " as a topology S if, so if every open set is open in S and in T. S is then called " coarser " than T. If the two topologies different, it is also said T was " strictly finer " than S and S is " real coarser " than T.

There are also generally topologies T and S, which can not be compared in this sense. For them there exists a unique common refinement, which is the coarsest topology on X, which comprises both topologies. Dual to this common refinement is given by the intersection topology. She is the finest topology that is included in both topologies. By the relation " is finer than " the topologies on a set to an association.

This way of speaking is compatible with the " fine " order of the environment systems as a filter: If x is a fixed point in space, then the generated by the finer topology T ambient filter V ( x) is finer than the U generated by the coarser topology S ( x ).

### Morphisms: Continuous pictures

As with any mathematical structure, there are also in the topological spaces structure -preserving mappings ( morphisms ). Here are the continuous maps: A picture is (globally) continuous if the inverse image of every open subset O of Y is an open set in X, formally: .

The isomorphisms here are called homeomorphisms, these are bijective continuous mappings whose inverse is also continuous. Structurally similar ( isomorphic ) topological spaces are called homeomorphic.

## Examples

- Exist on any ground set X as nontrivial examples of topologies: The indiscrete topology that contains only the empty set and the universal set. It is the coarsest topology on X.
- Discrete topology that includes all subsets. She is the finest topology on X.
- On an infinite set (eg the set of natural numbers ), one can introduce the kofinite topology: Open is the empty set and each subset of the complement of which contains only finitely many elements.
- Each strictly totally ordered set can be provided in a natural way with its order topology.
- Open spheres in a metric space generate (as base ) is a topology that induced by the topology metrics. Special metric spaces are normed spaces, here the metric and thus the natural topology ( norm topology ) is induced from the norm.

## Generation of topological spaces

- You can expand any system S of subsets of a set X to a topology on X by requiring that (at least) all amounts of S are open. Thus S is a sub-base for the topology on X.
- Each subset Y of a topological space X, a subspace topology are assigned. The open sets are precisely the intersections of the open sets in X with the subset Y.
- For each family of topological spaces the set-theoretic product of the basic quantities can be provided with the product topology: At finite products, the products of open sets of the factor spaces form a basis for this topology.
- For infinite products those products form of open sets of the factor spaces a base in which each comprise all but finitely many factors throughout the relevant space.
- If one chooses in an infinite product as a basis the cartesian products of open sets of the factor spaces, then you get the box topology on the product. This is ( i a genuine) finer than the product topology.