General topology

In mathematics, general 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 sets are those that can be covered by finitely many sets of arbitrarily small size.
Connected sets are 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.
Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

History

General topology grew out of a number of areas, most importantly the following:

the detailed study of subsets of the real line
the introduction of the manifold concept
the study of metric spaces, especially 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:

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 is called a topological space. The notation X_τ 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. A subset of X may be open, closed, both, or neither. The empty set and X itself are always both closed and open.

Basis for a topology

A base 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. 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 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.

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, 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 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.
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 linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges.
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.
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 and where a and b are elements of Γ.
Continuous functions

Continuity is expressed in terms of neighborhoods: is continuous at some point if and only if for any neighborhood of, there is a neighborhood of such that. Intuitively, continuity means no matter how "small" becomes, there is always a containing that maps inside and whose image under contains. This is equivalent to the condition that the preimages of the open sets in are open in. In metric spaces, this definition is equivalent to the ε-δ-definition that is often used in analysis.
An extreme example: if a set is given the discrete topology, all functions
to any topological space are continuous. On the other hand, if is equipped with the indiscrete topology and the space set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Neighborhood definition

Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of neighborhoods: f is continuous at some point x ∈ X if and only if for any neighborhood V of f, there is a neighborhood U of x such that f ⊆ V. Intuitively, continuity means no matter how "small" V becomes, there is always a U containing x that maps inside V.
If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.
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. At an isolated point, every function is continuous.