Topological space

Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in all branches of modern mathematics and can be seen as a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology.

Table of contents

1 Formal definition 2 Relations between topologies 3 Continuous functions 4 Alternative definitions 5 Examples of topological spaces 6 Constructing new topological spaces from given ones 7 Classification of topological spaces 7.1 Separation of points 7.2 Compactness 7.3 Countability conditions 7.4 Connectedness 7.5 Miscellaneous 8 Topological spaces with algebraic structure 9 History

Formal definition

Formally, a topological space is a set X together with a collection T of subsets of X (i.e., T is a subset of the power set of X) satisfying the following axioms:

1. The empty set and X are in T.
2. The union of any collection of sets in T is also in T.
3. The intersection of any pair of sets in T is also in T.

The set T is called a topology on X. The sets in T are referred to as open sets, and their complements in X are called closed sets. The elements of X are often called points. Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points.

Relations between topologies

A variety of useful and not-so-useful topologies can be placed on nearly any set to form a topological space. When every set in a topology T₁ is also found in a topology T₂, we say that T₂ is finer than T₁, and T₁ is coarser than T₂. A proof which relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. Other terms, such as stronger, weaker, larger, and smaller are used in literature but with little agreement on their meanings.

Continuous functions

A function between topological spaces is said to be continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "breaks" or "separations" in the function. A homeomorphism is a bijective mapping that is continuous and whose inverse is also continuous. Two spaces are said to be homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.

The category of all topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in all mathematics. The attempt to classify the objects of this category by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and K-theory, to name just a few.

Alternative definitions

There are many other equivalent ways to define a topological space. (In other words, each of the following defines a category equivalent to the category of topological spaces above.)

• Using de Morgan's laws, the axioms defining open sets become axioms defining closed sets:

The empty set and X are closed.

The intersection of any collection of closed sets is also closed.

The union of any pair of closed sets is also closed.

• The Kuratowski closure axioms determine the closed sets as the fixed points of an operator on the power set of X.

• A neighbourhood of a point x is any set that contains an open set containing x. The neighbourhood system at x consists of all neighbourhoods of x. A topology can be determined by a set of axioms concerning all neighbourhood systems. Equivalently, a topology can be determined by a nearness relation between sets and points.

• A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X the set of its accumulation points is specified.

Examples of topological spaces

• The set of real numbers R is a topological space: the open sets are generated by the base of open intervals. This means a set is open if it is the union of (possibly infinitely many) open intervalss. This is in many ways the most basic topological space and the one that guides most of our human intuition. However, relying on the real line as an intuitive guide for the general concept of topological space can often be dangerous.
• More generally, the Euclidean spaces Rⁿ are topological spaces, and the open sets are generated by open balls.
• Any metric space turns into a topological space if define the open sets to be generated by the set of all open balls. This includes useful infinite-dimensional spaces like Banach spaces and Hilbert spaces studied in functional analysis.
• The reals can also be given the upper-limit topology. Here, the open sets consist of the empty set, the whole real line, and all sets generated by half-open intervals of the form (a,b]. This topology on R is strictly larger than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from below in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.
• Every manifold is a topological space.
• Every simplex is a topological space. Simplexes are convex objects that are very useful in computational geometry. In 0, 1, 2 and 3 dimensional space the simplexes are the point, line segment, triangle and tetrahedron, respectively.
• Every simplicial complex is a topological space. A simplicial complex is made up of many simplices. Many geometric objects can be modeled by simplicial complexes -- see also Polytope.
• The Zariski topology is a purely algebraically defined topology 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 is a topological space that generalises many of the geometric aspects of graphss with vertices and edges.
• Many sets of 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 set can be given the discrete topology in which every set is open. The only convergent sequences or nets in this topology are those that are eventually constant.
• 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.
• Any infinite 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 the set.
• If Γ is an ordinal number, then the set [0, Γ] is a topological space, generated by the intervals (a,b], where a and b are elements of Γ.

Constructing new topological spaces from given ones

• 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 nonempty collection of topological spaces, the product can be given the product topology. For finite products, the open sets are generated by the products of open sets.
• A quotient space is defined as follows. If f: X → Y is a function and X is a topological space, then Y gets a topology where a set is open if and only if its inverse image is open. A common example comes from an equivalence relation defined on the topological space X: the map f is then the natural projection on the set of equivalence classes.
• The Vietoris topology on the set of all non-empty subsets of a topological space X is generated by the following basis: for every n-tuple U₁,....,U_n of open sets in X we construct a basis set consisting of all subsets of the union of the U_i which have non-empty intersection with each U_i.

Classification of topological spaces

Topological spaces can be broadly classified according to their degree of connectedness, their size, their degree of compactness and the degree of separation of their points and subsets. A great many terms are used in topology to achieve these distinctions. These terms and definitions are collected together in the Topology Glossary. Using these terms, we can give the following classification:

Separation of points

For a detailed treatment, see Separation axiom. Some of these terms are defined differently in older mathematical literature; see History of the separation axioms.

• Trivial topology. A space carries the trivial topology if all points are "lumped together" in the sense that there are only two open sets, the empty set and the whole space.
• T₀. A space is T₀ if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x.
• T₁. A space is T₁ if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T₀; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T₁ if all its singletons are closed. T₁ spaces are always T₀.
• Hausdorff or T₂. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. Hausdorff spaces are always T₁.
• Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
• Regular Hausdorff or T₃. A space is regular Hausdorff if it is a regular T₀ space. (A regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.)
• Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated.
• Tychonoff, Completely regular Hausdorff, Completely T₃ or T_3½. A Tychonoff space is a completely regular T₀ space. (A completely regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
• Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.
• Normal Hausdorff or T₄. A normal space is Hausdorff if and only if it is T₁. Normal Hausdorff spaces are always Tychonoff.
• Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.
• Completely normal Hausdorff or T₅. A completely normal space is Hausdorff if and only if it is T₁. Completely normal Hausdorff spaces are always normal Hausdorff.
• Discrete space. A space is discrete if all of its points are completely isolated, i.e. if any subset is open.

Compactness

• Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.
• Lindelöf. A space is Lindelöf if every open cover has a countable subcover.
• Countably compact. A space is countably compact if every countable open cover has a finite subcover.
• Compact. A space is compact if every open cover has a finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.
• Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Locally compact Hausdorff spaces are always Tychonoff.

Countability conditions

• Separable. A space is separable if it has a countable dense subset.
• First-countable. A space is first-countable if every point has a countable local base.
• Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

Connectedness

• Connected. A space X is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.
• Locally connected. A space is locally connected if every point has a local base consisting of connected sets.
• Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.
• Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.
• Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
• Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S¹ → X is homotopic to a constant map.
• Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.

Miscellaneous

• Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.
• Polish. A space is called Polish if it is metrizable with a separable and complete metric.
• Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.
• Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.
• Homogeneous. A space X is homogeneous if for every x and y in X there is a homeomorphism f : X -> X such that f(x) = y. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.

Topological spaces with algebraic structure

It is almost universally true that all "large" algebraic objects carry a natural topology which is compatible with the algebraic operations. In order to study these objects, one typically has to take the topology into account. This leads to concepts such as topological groups, topological vector spaces and topological rings.

History

See topology.