Topological space

Examples and counterexamples for topologies - the six figures represent subsets of the power set of {1,2,3} (the small circle at the top left is the empty set). The first four are topologies; in the example at the bottom left {2,3} is missing, at the bottom right {2} for the topology property.

A topological space is the fundamental subject of the sub-discipline topology of mathematics . By introducing a topological structure on a set, intuitive positional relationships such as “proximity” and “striving against” can be transferred from the visual space to a large number of very general structures and given a precise meaning.

definition

A topology is a system of sets consisting of subsets of a basic set , called open or open sets , which satisfy the following axioms : ${\displaystyle T}$ ${\displaystyle X}$

• The empty set and the basic set are open. ${\displaystyle X}$
• The intersection of finitely many open sets is open. (Suffice it to say that the intersection of two open sets is open.)
• The union of any number of open sets is open.

Then called a topology on , and the couple a topological space . ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle (X,T)}$

Basic concepts

Speaking: elements are points, the set is a space

From the visual space, the designation “point” for the elements of the basic set and the designation “(topological) space” for the set that carries the topological structure have prevailed. A topological space is formally correct but the pair of the structure-bearing set and the structure-defining system (the “topology”) of subsets. ${\displaystyle X}$ ${\displaystyle (X,T)}$ ${\displaystyle X}$ ${\displaystyle T}$

Dual: completed

A subset of topological space whose complement is an open set is called closed . If one dualises the definition formulated above and replaces the word “open” with “closed” (as well as interchanging intersection and union), the result is an equivalent definition of the term “topological space” using its system of closed sets. ${\displaystyle X}$

Environments

In a topological space, each point has a filter of surroundings . This allows the intuitive concept of “proximity” to be understood mathematically. This term can also be used as a basis for a definition of topological space. ${\displaystyle x}$ ${\displaystyle U(x)}$

Comparison of topologies: coarser and finer

→ Main article : Coarser and finer topologies

On a fixed amount can be certain topologies and compare: It's called a topology finer than a topology when , ie when each in the open set in is open. then means coarser than . If the two topologies are different, it is also said that it is really finer than and is really coarser than . ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle S\subseteq T}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle T}$

There are generally topologies and that cannot be compared in this sense. There is a clear common refinement for them , this is the coarsest topology that includes both topologies. The topology given by the intersection is dual to this common refinement . It is the finest topology contained in both topologies. With the relation “is finer than” the topologies on a set become a group . ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle S\cap T}$

This way of speaking is compatible with the "finer" order of the surrounding systems as a filter: If there is a fixed point in space, then the environment filter generated by the finer topology is finer than the one generated by the coarser topology . ${\displaystyle x}$ ${\displaystyle T}$ ${\displaystyle V(x)}$ ${\displaystyle S}$ ${\displaystyle U(x)}$

Morphisms: Continuous mappings

As with every mathematical structure, there are also structure-preserving images ( morphisms ) in the topological spaces . Here are the continuous maps : A picture is (global) steadily when the archetype of every open subset of an open set in is formal: . ${\displaystyle f\colon (X,S)\to (Y,T)}$ ${\displaystyle O}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle O\in T\implies f^{-1}(O)\in S}$

The isomorphisms are called homeomorphisms here ; these are bijective continuous mappings, the inversion of which is also continuous. Structurally similar (isomorphic) topological spaces are called homeomorphic.

Examples

Topological spaces in relation to other structures defining proximity

• On each base set there are trivial examples of topologies: ${\displaystyle X}$
  1. The indiscreet topology that only contains the empty set and the basic set. It is the coarsest topology . ${\displaystyle X}$
  2. The discrete topology that contains all subsets. It's the finest topology on . ${\displaystyle X}$
• The cofinite topology can be introduced on an infinite set (e.g. the set of natural numbers) : Open is the empty set as well as every subset of whose complement only contains finitely many elements. ${\displaystyle M}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle M}$
• Every strictly total ordered set can be given its order topology in a natural way .
• The open spheres in a metric space create (as a basis ) a topology, the topology induced by the metric.
  • Special metric spaces are the standardized spaces , here the metric and thus the natural topology ( standard topology ) is induced by the standard .
• Some concrete topological spaces with specially constructed properties bear the names of their discoverers, e.g. B. Arens-Fort room , Cantor room , Hilbert cube , Michael straight , Niemytzki room , Sorgenfrey level , Tichonow plank etc.

Generation of topological spaces

• Any system of subsets of a set can be extended to a topology by requiring that (at least) all sets are open. This becomes the sub-basis of a topology . ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle X}$
• A subspace topology can be assigned to each subset of a topological space . The open sets are precisely the intersections of the open sets with the subset . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle Y}$
• For every family of topological spaces, the set-theoretical product of the basic sets can be given the product topology :
  • In the case of finite products, the products of the open sets from the factor spaces form a basis of this topology.
  • In the case of infinite products, those products of open sets from the factor spaces form a basis in which all but a finite number of factors each encompass the entire space concerned.
  • If you choose the Cartesian products of open sets from the factor spaces as a basis in an infinite product, you get the box topology on the product. This is (generally really) finer than the product topology.
• A generalization of the subspace and product topology examples is the construction of an initial topology . Here the topology is defined on a set by the requirement that certain mappings from into other topological spaces should be continuous. The initial topology is the coarsest topology with this property. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$
• A quotient topology is created by gluing (identifying) certain points together in a topological space . Formally, this is done through an equivalence relation , so the points of the quotient space are classes of points . ${\displaystyle X}$ ${\displaystyle X}$
• A generalization of the quotient topology example is the construction of a final topology . Here the topology is defined on a set by the requirement that certain mappings from other topological spaces should be continuous. The final topology is the finest topology with this property. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

literature

• Gerhard Preuss: General Topology . 2nd corrected edition. Springer, Berlin et al. 1975, ISBN 3-540-07427-9 , ( university text ).
• Horst Schubert: Topology. An introduction. 4th edition. Teubner, Stuttgart 1975, ISBN 3-519-12200-6 , ( Mathematical Guidelines ).
• Klaus Jänich : Topology. 6th edition. Springer, Berlin et al. 1999, ISBN 3-540-65361-9 , ( Springer textbook ).
• Charles E. Aull, Robert Lowen (Eds.): Handbook of the History of General Topology . Volume 3. Kluwer Academic, Dordrecht 2001, ISBN 0-7923-6970-X .
• Boto von Querenburg : Set theoretical topology. 3rd revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 , ( Springer textbook ).
• René Bartsch: General Topology I . Oldenbourg, Munich et al. 2007, ISBN 978-3-486-58158-4 .