Topological space

Short description: Mathematical space with a notion of closeness

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.

A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness.[1][2] Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called general topology (or point-set topology).

## Contents

* 1 History
* 2 Examples of topological spaces
* 2.1 Topology from other topologies
* 2.2 Metric spaces
* 2.3 Topology from algebraic structure
* 2.4 Topological spaces with order structure
* 2.5 Topology from other structure
* 3 Classification of topological spaces
* 4 See also
* 5 Citations
* 6 Bibliography
* 7 External links

## History

A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A.

Carl Friedrich Gauss, Gauss 1827

Around 1735, Leonhard Euler discovered the formula V−E+F=2 relating the number of vertices (V), edges (E) and faces (F) of a convex polyhedron, and hence of a planar graph. The study and generalization of this formula, specifically by Cauchy (1789–1857) and L'Huilier (1750–1840), boosted the study of topology. In 1827, Carl Friedrich Gauss published General investigations of curved surfaces, which in section 3 defines the curved surface in a similar manner to the modern topological understanding.

Yet, "until Riemann's work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered".[3] "Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not."[3]

The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894.[4] In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.

Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory". Metric spaces had been defined earlier in 1906 by Maurice Fréchet, though it was Hausdorff who popularised the term "metric space" (German: metrischer Raum).[5]Cite error: Closing `</ref>` missing for `<ref>` tag

In category theory, one of the fundamental categories is Top, which denotes the category of topological spaces whose objects are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory.[citation needed]

## Examples of topological spaces

See also: List of topologies

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique.

There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.

Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T1 topology on any infinite set.[6]

Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.

The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a,b). This topology on ℝ is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.

If γ is an ordinal number, then the set γ=[0,γ) may be endowed with the order topology generated by the intervals (α,β), [0,β), and (α,γ) where α and β are elements of γ.

Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from .

The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.

### Topology from other topologies

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Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. This construction is a special case of an initial topology.

A quotient space is defined as follows: if X is a topological space and Y is a set, and if f:X->Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes. This construction is a special case of a final topology.

### Metric spaces

Main article: Metric space

Metric spaces embody a metric, a precise notion of distance between points.

Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

There are many ways of defining a topology on ℝ, the set of real numbers. The standard topology on ℝ is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces ℝn can be given a topology. In the usual topology on ℝn the basic open sets are the open balls. Similarly, ℂ, the set of complex numbers, and ℂn have a standard topology in which the basic open sets are open balls.

### Topology from algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups, topological rings, topological fields and topological vector spaces over the latter. Local fields are topological fields important in number theory.

The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On ℝn or ℂn, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.

### Topological spaces with order structure

* Spectral: A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).
* Specialization preorder: In a space the specialization preorder (or canonical preorder) is defined by x≤y if and only if cl⁡{x}⊆cl⁡{y}, where cl denotes an operator satisfying the Kuratowski closure axioms.

### Topology from other structure

If Γ is a filter on a set X then {∅}∪Γ is a topology on X.

Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

A linear graph has a natural topology that generalizes many of the geometric aspects of graphs with vertices and edges.

Outer space of a free group Fn consists of the so-called "marked metric graph structures" of volume 1 on Fn.[7]

## Classification of topological spaces

Main article: Topological property

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms. For algebraic invariants see algebraic topology.

## See also

* Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.
* Compact space - Type of mathematical space
* Convergence space - Generalization of the notion of convergence that is found in general topology
* Exterior space
* Hausdorff space - Type of topological space
* Hilbert space - Type of vector space in math
* Hemicontinuity - Semicontinuity for set-valued functions
* Linear subspace - In mathematics, vector subspace
* Pointless topology
* Quasitopological space - Function in topology
* Relatively compact subspace - Subset of a topological space whose closure is compact
* Space (mathematics) - Mathematical set with some added structure

## Citations

1. ↑ Schubert 1968, p. 13
2. ↑ Sutherland, W. A. (1975). Introduction to metric and topological spaces. Oxford [England]: Clarendon Press. ISBN 0-19-853155-9. OCLC 1679102.
3. ↑ 3.0 3.1 Gallier & Xu 2013.
4. ↑ J. Stillwell, Mathematics and its history
5. ↑ metric space (3rd ed.), Oxford University Press, September 2005, http://oed.com/search?searchType=dictionary&q=metric+space (Subscription or UK public library membership required.)
6. ↑ Anderson, B. A.; Stewart, D. G. (1969). "T1-complements of T1 topologies". Proceedings of the American Mathematical Society 23: 77–81. doi:10.2307/2037491.
7. ↑ Culler, Marc; Vogtmann, Karen (1986). "Moduli of graphs and automorphisms of free groups". Inventiones Mathematicae 84 (1): 91–119. doi:10.1007/BF01388734. Bibcode: 1986InMat..84...91C. https://www.math.cornell.edu/~vogtmann/ScannedPapers/1986.0084.pdf.

## Bibliography

* Armstrong, M. A. (1983). Basic Topology. Undergraduate Texts in Mathematics. Springer. ISBN 0-387-90839-0.
* Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997). ISBN 0-387-97926-3.
* Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).
* Brown, Ronald (2006). Topology and Groupoids. Booksurge. ISBN 1-4196-2722-8. http://groupoids.org.uk/topgpds.html. (3rd edition of differently titled books)
* Čech, Eduard; Point Sets, Academic Press (1969).
* Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997). ISBN 0-387-94327-7.
* Gallier, Jean; Xu, Dianna (2013). A Guide to the Classification Theorem for Compact Surfaces. Springer.
* Gauss, Carl Friedrich (1827). General investigations of curved surfaces.
* Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0-07-037988-2.
* Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
* Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.
* Schubert, Horst (1968), Topology, Macdonald Technical & Scientific, ISBN 0-356-02077-0
* Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). ISBN 0-03-079485-4.
* Vaidyanathaswamy, R. (1999). Set Topology. Chelsea Publishing Co.. ISBN 0486404560.
* Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6. https://books.google.com/books?id=-o8xJQ7Ag2cC&q="topological+space".

## External links

* Hazewinkel, Michiel, ed. (2001), "Topological space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/t093130

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