Topology is the study of properties of geometric spaces which are preserved by continuous deformations (intuitively, stretching, rotating, or bending are continuous deformations; tearing or gluing are not). The theory originated as a way to classify and study properties of shapes in but the axioms of what is now known as point-set topology have proved to be both rich enough and general enough to apply the ideas of abstract topology to many other areas of mathematics. For instance, topological ideas arise naturally in calculus and analysis, but they are also quite important in modern number theory. The basic structure of topology is an axiomatic way to make sense of when two points in a set are "near" each other. In down-to-earth situations such as Euclidean space, there is a standard Euclidean distance function that measures how far apart two points are. This turns into a metric space, which is the most natural example of a topological space. Concepts such as continuity and limit can be defined in terms of this distance function (or metric).
More generally, even when there is not a metric, the main concepts of topology can be defined by replacing the metric by a notion of open set. In a metric space, a set is open if each point in the set has a neighborhood contained entirely in the set. A neighborhood of a point in a metric space is the set of all points within a certain distance of
When there is no well-defined distance function, the more abstract definition proceeds instead by specifying directly which subsets are open. Under certain necessary assumptions on this collection of open sets (called a topology), it is possible to meaningfully extend the definitions of important concepts such as continuity, connectedness, and compactness to these more abstract topological spaces.
As mentioned in the introduction, the general notion of a topology on a set is the foundation of the theory.
Let be a set. A topology on is a collection of subsets of satisfying the following properties:
and are both in
The union of a collection of sets in is in
The intersection of finitely many sets in is in
The subsets in are called open sets.
Every set has two basic (but important) topologies. The discrete topology on is the topology consisting of every subset of ("every subset is open"). The indiscrete topology on is the topology such that the only two subsets in are and ("no nonempty proper subset is open"). Both of these are clearly topologies (they satisfy the conditions in the definition).
The discrete topology is the only topology in which every one-point subset of is open: if every one-point subset is open, every subset can be expressed as a union of one-point subsets, so it is open by axiom 2.
Main article: Metric space
The most natural examples of topological spaces arise from a metric, which is a function that assigns a nonnegative real number distance to any two points in the space. The conditions that define a metric are
- (the triangle inequality: the shortest distance between two points is a straight line).
The topology induced by a metric contains open sets with the property that, for any point there is some such that all points with are also in That is, any point in has a "ball of radius " around it, which is completely contained inside
Topologies induced by metric spaces have many special properties that arbitrary topologies do not have. One useful example is the Hausdorff property: any two distinct points and can be separated by open sets. That is, there are open sets with To see this, suppose ; then let and be the balls of radius around and respectively. The fact that and are disjoint follows from the triangle inequality.
A topology whose open sets are the same as the open sets induced by some metric is called metrizable. A non-Hausdorff topology (like the cofinite topology on ) is non-metrizable.
Let be a set. A basis for is a collection of subsets of such that
- For any two sets and point there is a third set such that and
Let be a metric space. The collection of balls where is a base.
The two properties of a base give the following theorem, whose proof is left as an exercise:
The collection of sets which are unions of elements in a base for forms a topology on This is often written " is a base for "
The collection of balls in a metric space is a base for the standard topology coming from the metric.
Let be the collection of one-point sets of Then is a base for the discrete topology on
There are several ways to make new topological spaces from old ones. Here are two common ones.
Let be a topological space and a subset of The subspace topology on is the topology whose open sets are the sets where is an open set in
It is easy to check that this construction always gives a valid topology on Note that this introduces some ambiguity about open and closed sets. Saying that a subset of, say, is open is ambiguous without context: it might be open in some subspace topology but not open inside
Find subsets of such that is open in with the subspace topology, but is not open in
One trivial example is is not open in but since and is open in is open in
The interval is also open in since it is the intersection
The second construction involves the Cartesian product of two topological spaces. Recall that the product of two sets consists of ordered pairs with
Let be topological spaces. Then the product topology on consists of open sets which are unions of sets where is open in and is open in
So the product topology is the topology whose base consists of subsets with open in and open in
Note that the open sets are not all of the form It is necessary to take unions, since the union of and is not necessarily of the form
Show that the product topology on is the same as the standard Euclidean topology on
The idea is that the base for the open sets of the product topology is the product of open intervals, which is an open rectangle. The base for the open sets of the standard Euclidean topology is an open disk. The two topologies give the same open sets, because inside any open disk around a point there is an open rectangle, and inside any open rectangle around there is an open disk. So a set that is open in one topology is open in the other.
The product topology on infinite Cartesian products is more subtle: the product topology on a product has a base consisting of products where all but finitely many of the are equal to
The topology whose base consists of products is called the box topology, but it is not the "correct" topology for many applications--there are too many open sets. In particular, the product of compact spaces might not be compact under the box topology, but it is under the product topology.
Once a mathematical object has been defined, the next step is to specify what maps between the objects look like. The appropriate functions to consider between topological spaces are the continuous functions, and the definition of a continuous function is quite concise:
A function where are topological spaces, is continuous if, for every open set is open in
Main page: Homeomorphism
Equipped with the definition of continuity given above, one can define what it means for two topological spaces to be the same. Essentially, it means that there should be a bijective map between them that is also a bijection on the open sets. Such a map is called a homeomorphism, and two homeomorphic topological spaces are considered to be the same. The formal definition is as follows:
A function between topological spaces is called a homeomorphism if is continuous and a bijection, and is also continuous.
For an important example of a non-obvious homeomorphism, see stereographic projection: a (two-dimensional) sphere minus a point is homeomorphic to the plane.
There are many important properties which can be used to characterize topological spaces. Two of the most important are connectedness and compactness. Since they are both preserved by continuous functions--i.e. the continuous image of a connected space is connected, and the continuous image of a compact space is compact--these properties remain invariant under homeomorphism. Properties which are invariant under homeomorphism are called topological properties, and they are useful tools in the classification of topological spaces.
Other examples of topological properties include