General topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
The fundamental concepts in point-set topology are continuity, compactness, and connectedness:
- Continuous functions, intuitively, take nearby points to nearby points.
- Compact setsare those that can be covered by finitely many sets of arbitrarily small size.
- Connected setsare sets that cannot be divided into two pieces that are far apart.
The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.
History
General topology grew out of a number of areas, most importantly the following:
- the detailed study of subsets of the real line(once known as the topology of point sets; this usage is now obsolete)
- the introduction of the manifold concept
- the study of normed linear spaces, in the early days of functional analysis.
General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.
A topology on a set
Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:[1][2]
- Both the empty set and X are elements of τ
- Any union of elements of τ is an element of τ
- Any intersection of finitely many elements of τ is an element of τ
If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ.
The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open.
Basis for a topology
A base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.[3][4] We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.
Subspace and quotient
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.
A
Examples of topological spaces
A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.
Discrete and trivial topologies
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.
Cofinite and cocountable topologies
Any set can be given the
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.
Topologies on the real and complex numbers
There are many ways to define 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.
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.
The metric topology
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.
Further examples
- 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.
- Every manifold has a natural topology, since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn.
- 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.
- A .
- Many sets of linear operators in functional analysisare endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
- Any local field has a topology native to it, and this can be extended to vector spaces over that field.
- The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.
- If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ.
Continuous functions
Continuity is expressed in terms of
An extreme example: if a set X is given the
to any topological space T are continuous. On the other hand, if X is equipped with the
Alternative definitions
Several
Neighborhood definition
Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of
If X and Y are metric spaces, it is equivalent to consider the
Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.
Sequences and nets
In several contexts, the topology of a space is conveniently specified in terms of
In detail, a function f: X → Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x).[6] Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.
Closure operator definition
Instead of specifying the open subsets of a topological space, the topology can also be determined by a
between topological spaces is continuous in the sense above if and only if for all subsets A of X
That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A). This is equivalent to the requirement that for all subsets A' of X'
Moreover,
is continuous if and only if
for any subset A of X.
Properties
If f: X → Y and g: Y → Z are continuous, then so is the composition g ∘ f: X → Z. If f: X → Y is continuous and
- X is compact, then f(X) is compact.
- X is connected, then f(X) is connected.
- X is path-connected, then f(X) is path-connected.
- X is Lindelöf, then f(X) is Lindelöf.
- X is separable, then f(X) is separable.
The possible topologies on a fixed set X are
- idX: (X, τ2) → (X, τ1)
is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function
stays continuous if the topology τY is replaced by a coarser topology and/or τX is replaced by a finer topology.
Homeomorphisms
Symmetric to the concept of a continuous map is an
If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.
Defining topologies via continuous functions
Given a function
where X is a topological space and S is a set (without a specified topology), the
Dually, for a function f from a set S to a topological space, the initial topology on S has as open subsets A of S those subsets for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.
A topology on a set S is uniquely determined by the class of all continuous functions into all topological spaces X. Dually, a similar idea can be applied to maps
Compact sets
Formally, a
of open subsets of X such that
there is a finite subset J of A such that
Some branches of mathematics such as
Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).
Every continuous image of a compact space is compact.
A compact subset of a Hausdorff space is closed.
Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
Every sequence of points in a compact metric space has a convergent subsequence.
Every compact finite-dimensional manifold can be embedded in some Euclidean space Rn.
Connected sets
A
For a topological space X the following conditions are equivalent:
- X is connected.
- X cannot be divided into two disjoint nonempty closed sets.
- The only subsets of X that are both open and closed (clopen sets) are X and the empty set.
- The only subsets of X with empty boundary are X and the empty set.
- X cannot be written as the union of two nonempty separated sets.
- The only continuous functions from X to {0,1}, the two-point space endowed with the discrete topology, are constant.
Every interval in R is connected.
The continuous image of a connected space is connected.
Connected components
The
Let be the connected component of x in a topological space X, and be the intersection of all open-closed sets containing x (called quasi-component of x.) Then where the equality holds if X is compact Hausdorff or locally connected.
Disconnected spaces
A space in which all components are one-point sets is called
Path-connected sets
A
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line L* and the topologist's sine curve.
However, subsets of the
Products of spaces
Given X such that
is the Cartesian product of the topological spaces Xi, indexed by , and the
The open sets in the product topology are unions (finite or infinite) of sets of the form , where each Ui is open in Xi and Ui ≠ Xi only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the Xi gives a basis for the product .
The product topology on X is the topology generated by sets of the form pi−1(U), where i is in I and U is an open subset of Xi. In other words, the sets {pi−1(U)} form a
In general, the product of the topologies of each Xi forms a basis for what is called the
Related to compactness is Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
Separation axioms
Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.
Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.
In all of the following definitions, X is again a topological space.
- X is topologically distinguishable. (It is a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)
- X is T1, or accessible or Fréchet, if any two distinct points in X are separated. Thus, X is T1 if and only if it is both T0 and R0. (Though you may say such things as T1 space, Fréchet topology, and Suppose that the topological space X is Fréchet, avoid saying Fréchet space in this context, since there is another entirely different notion of Fréchet space in functional analysis.)
- X is Hausdorff, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1.
- X is T2½, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T2½ space must also be Hausdorff.
- X is regular, or T3, if it is T0 and if given any point x and closed set F in X such that x does not belong to F, they are separated by neighbourhoods. (In fact, in a regular space, any such x and F is also separated by closed neighbourhoods.)
- X is Tychonoff, or T3½, completely T3, or completely regular, if it is T0 and if f, given any point x and closed set F in X such that x does not belong to F, they are separated by a continuous function.
- X is normal, or T4, if it is Hausdorff and if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.)
- X is completely normal, or T5 or completely T4, if it is T1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.
- X is perfectly normal, or T6 or perfectly T4, if it is T1 and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
Countability axioms
An axiom of countability is a
Important countability axioms for topological spaces:
- sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set
- first-countable space: every point has a countable neighbourhood basis (local base)
- second-countable space: the topology has a countable base
- dense subspace
- open coverhas a countable subcover
- σ-compact space: there exists a countable cover by compact spaces
Relations:
- Every first countable space is sequential.
- Every second-countable space is first-countable, separable, and Lindelöf.
- Every σ-compact space is Lindelöf.
- A metric space is first-countable.
- For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.
Metric spaces
A metric space[7] is an ordered pair where is a set and is a
such that for any , the following holds:
- (non-negative),
- iff (identity of indiscernibles),
- (symmetry) and
- (triangle inequality) .
The function is also called distance function or simply distance. Often, is omitted and one just writes for a metric space if it is clear from the context what metric is used.
Every
.The
Baire category theorem
The
Any open subspace of a Baire space is itself a Baire space.
Main areas of research
Continuum theory
A continuum (pl continua) is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and analysis, and their properties are strong enough to yield many 'geometric' features.
Dynamical systems
Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include fluid dynamics, billiards and flows on manifolds. The topological characteristics of fractals in fractal geometry, of Julia sets and the Mandelbrot set arising in complex dynamics, and of attractors in differential equations are often critical to understanding these systems.[citation needed]
Pointless topology
Pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann.[9] The ideas of pointless topology are closely related to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.
Dimension theory
Dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.
Topological algebras
A topological algebra A over a
that makes it an algebra over K. A unital associative topological algebra is a topological ring.
The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).
Metrizability theory
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric
such that the topology induced by d is . Metrization theorems are
Set-theoretic topology
Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC). A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.
See also
- List of examples in general topology
- Glossary of general topologyfor detailed definitions
- List of general topology topics for related articles
- Category of topological spaces
References
- ^ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
- ^ Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
- ISBN 0-471-83817-9. Retrieved 27 July 2012.
Definition. A collection B of subsets of a topological space (X,T) is called a basis for T if every open set can be expressed as a union of members of B.
- ISBN 0-387-90839-0. Retrieved 13 June 2013.
Suppose we have a topology on a set X, and a collection of open sets such that every open set is a union of members of . Then is called a base for the topology...
- JSTOR 2370388.
- ^ Heine, E. (1872). "Die Elemente der Functionenlehre." Journal für die reine und angewandte Mathematik. 74: 172–188.
- Maurice Fréchetintroduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22 (1906) 1–74.
- ^ R. Baire. Sur les fonctions de variables réelles. Ann. di Mat., 3:1–123, 1899.
- ^ Garrett Birkhoff, VON NEUMANN AND LATTICE THEORY, John Von Neumann 1903-1957, J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5
Further reading
Some standard books on general topology include:
- ISBN 0-387-19374-X.
- John L. Kelley (1955) General Topology, link from Internet Archive, originally published by David Van Nostrand Company.
- ISBN 0-486-43479-6.
- ISBN 0-13-181629-2.
- ISBN 1-575-24238-9.
- ISBN 0-470-09605-5.
- ISBN 3-88538-006-4.
- MR 0507446
- O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, Elementary Topology: Textbook in Problems, ISBN 978-0-8218-4506-6.
The arXiv subject code is math.GN.
External links
- Media related to General topology at Wikimedia Commons