1 Definition and basic ideas

A topological space is a set together with a chosen family of subsets, called open sets, that encodes a notion of geometric proximity without using distances. The framework is broad enough to describe familiar spaces from analysis and geometry, as well as highly abstract constructions in algebra and logic. Once a topology is specified, one can define continuous maps, limits, connectedness, compactness, and many other global properties.

1.1 Sets and subsets

The underlying set of a topological space may consist of points of any kind: numbers, geometric objects, functions, or even more abstract elements. The topology does not alter the set itself; it adds structure by selecting which subsets are to be regarded as open. Different topologies on the same set can produce very different notions of continuity and convergence.

1.2 Open sets

Open sets are the primitive building blocks of topology. Intuitively, an open set is a region that does not include its boundary points, though this picture is only reliable in spaces that come from familiar geometric settings. In the abstract theory, openness is determined by axioms rather than by distance.

1.2.1 Topology axioms

A collection of subsets of a set forms a topology if it contains the empty set and the whole set, is closed under arbitrary unions, and is closed under finite intersections. These conditions are designed to make open sets behave well under natural operations. A set equipped with such a collection is called a topological space.

1.2.2 Neighborhoods

A neighborhood of a point is a set that contains an open set around that point. Neighborhoods provide a flexible way to describe local behavior, especially when discussing continuity and convergence. In many contexts, a neighborhood is not required to be open itself, only to include some open region surrounding the point.

1.3 Closed sets

Closed sets are complements of open sets. They are useful for describing boundaries, limit points, and compactness. Many important results can be stated either in terms of open sets or in terms of closed sets, and the two viewpoints are often interchangeable.

1.4 Basis and subbasis

A basis is a collection of open sets from which every open set can be built as a union. A subbasis is an even smaller family whose finite intersections generate a basis. These notions are convenient because they allow a topology to be specified economically, especially in product spaces and other constructions.

2 Examples of topological spaces

Examples help show that the axioms are flexible and widely applicable. Some topologies are extremely fine, while others are very coarse. Many standard spaces in mathematics arise naturally from these basic examples.

2.1 Discrete topology

In the discrete topology, every subset is open. This is the finest possible topology on a set and makes every function out of the space continuous. It is often used as a simple test case because its local structure is trivial.

2.2 Indiscrete topology

The indiscrete topology has only two open sets: the empty set and the entire space. It is the coarsest possible topology. In such a space, continuity conditions become weak, but separation properties are minimal.

2.3 Metric spaces as topological spaces

Any metric space gives rise to a topology by declaring open sets to be those that contain, around each point, a small open ball. This construction connects topology with distance-based geometry and analysis. Many familiar spaces, such as Euclidean spaces, are studied first as metric spaces and then as topological spaces.

2.4 Subspace topology

If a subset is taken from a topological space, it inherits a natural topology by intersecting the ambient open sets with the subset. This is called the subspace topology. It allows smaller spaces to be studied as parts of larger ones without losing topological structure.

2.5 Product topology

The product topology is defined on a Cartesian product of spaces so that projection maps are continuous. It is the standard topology for combining spaces into a larger one. Basic open sets in a product topology typically constrain only finitely many coordinates.

2.6 Quotient topology

A quotient topology is formed by identifying points through an equivalence relation or by collapsing parts of a space to single points. It is the natural topology that makes a given surjective map continuous and open-set structure compatible with the identification. Quotient spaces are central in many geometric constructions.

3 Continuous maps

Continuous maps preserve topological structure in the same broad sense that smooth maps preserve differentiable structure. They are the morphisms of topology and provide the main means of comparing spaces.

3.1 Definition of continuity

A function between topological spaces is continuous if the preimage of every open set is open. This definition avoids coordinates and distances, yet matches the classical notion of continuity in metric spaces. It also behaves well under composition.

3.2 Homeomorphisms

A homeomorphism is a continuous bijection whose inverse is also continuous. Two spaces related by a homeomorphism are considered topologically equivalent. Homeomorphism captures the idea that spaces have the same shape from the viewpoint of topology, even if they may differ in geometry or presentation.

3.3 Initial and final topologies

The initial topology on a set is the coarsest topology making a chosen family of maps into the set continuous. The final topology is the finest topology making a chosen family of maps out of the set continuous. These constructions organize many common examples and clarify how topologies are induced by maps.

4 Topological properties

Topological properties are features preserved by homeomorphisms. They describe the large-scale structure of spaces and often determine what kinds of theorems apply.

4.1 Separation axioms

Separation axioms measure how well distinct points and closed sets can be distinguished by open sets. They form a hierarchy of increasingly strong conditions. These axioms are especially important in general topology and analysis.

4.1.1 T0, T1, and T2 spaces

A T0 space allows any two distinct points to be topologically distinguishable. A T1 space requires that each point be closed. A T2 space, also called a Hausdorff space, requires that distinct points have disjoint neighborhoods. These conditions are common assumptions in many areas of mathematics.

4.1.2 Normal and regular spaces

Regular spaces allow points to be separated from closed sets by neighborhoods, while normal spaces allow two disjoint closed sets to be separated in a stronger way. Such properties are important in extension theorems and in the study of functions on spaces. They often support arguments that depend on controlled separation.

4.2 Compactness

Compactness is a global finiteness condition that generalizes the behavior of closed and bounded subsets of Euclidean space. It often allows local information to be combined into a global conclusion. Compact spaces frequently have strong structural properties.

4.2.1 Open covers and subcovers

A space is compact if every open cover has a finite subcover. This means that whenever open sets cover the whole space, only finitely many are needed to do the job. The notion is fundamental in analysis, where it supports many existence and convergence results.

4.2.2 Local compactness

A space is locally compact if each point has a neighborhood with compact closure, or an equivalent local compactness property depending on the setting. This condition combines local manageability with useful global behavior. It appears in harmonic analysis, geometry, and the theory of topological groups.

4.3 Connectedness

Connectedness expresses the idea that a space is in one piece. It rules out decompositions into two separated nonempty open sets. This property is central to studying continuity and the overall shape of a space.

4.3.1 Path connectedness

A space is path connected if any two points can be joined by a continuous path. Path connectedness is stronger than connectedness in general, though the two notions coincide in many familiar settings. It gives a more intuitive picture of being able to move continuously within a space.

4.3.2 Components

Connected components are maximal connected subspaces. They partition a space into its connected pieces. In a similar way, path components partition a space into path connected pieces, when paths are available.

4.4 Countability conditions

Countability conditions control how complicated the local or global topology can be. They are useful in analysis because they often make spaces more manageable and connect topology with sequences and countable constructions.

4.4.1 First countability

A space is first countable if each point has a countable local basis of neighborhoods. This property helps sequences capture local behavior more effectively. Many standard spaces are first countable, including metric spaces.

4.4.2 Second countability

A space is second countable if it has a countable basis for its topology. This is a stronger condition with significant consequences, including manageable descriptions of open sets and favorable separation and compactness behavior in many contexts.

5 Constructions and operations

Topological spaces are often built from simpler ones using standard operations. These constructions allow new examples to be formed while preserving or modifying desired properties.

5.1 Subspaces

Subspaces arise by restricting attention to a subset of a space with the induced topology. This operation is one of the most basic in topology. It preserves the ambient structure as much as possible while focusing on a region of interest.

5.2 Products

Products combine multiple spaces into a single space whose points are tuples. The product topology ensures that each coordinate projection is continuous. This construction is essential in studying families of spaces simultaneously.

5.3 Coproducts and disjoint unions

Coproducts, often realized as disjoint unions, combine spaces without identifying points from different pieces. Each component retains its own topology, and the resulting space keeps the pieces separate. This is useful when a theory needs a space made from distinct parts.

5.4 Quotients

Quotients form new spaces by collapsing subsets or identifying points according to an equivalence relation. The topology is chosen so that the identification map behaves continuously. Quotient constructions are widely used in geometry, algebraic topology, and manifold theory.

5.5 Topological sums

A topological sum is another name for a disjoint union equipped with the natural topology from its components. It is a standard way to assemble a space from a family of spaces indexed by any set. Each part sits openly inside the whole.

6 Convergence and limit concepts

Topology generalizes the idea of convergence beyond distances. It provides several related tools for describing when objects approach one another and when points are limit points of a set.

6.1 Sequences

Sequences are ordered lists of points that may converge to a limit. In first countable spaces, sequences often capture much of the local topology. However, in more general spaces, sequences may be insufficient to describe all convergence phenomena.

6.2 Nets

Nets generalize sequences by indexing points with directed sets rather than the natural numbers. They are powerful enough to characterize convergence in arbitrary topological spaces. Nets are especially useful when countability assumptions fail.

6.3 Filters

Filters describe convergence through collections of sets that become progressively smaller or more refined. They provide an alternative language to nets and are often convenient in abstract proofs. Filters are closely related to neighborhoods and cluster points.

6.4 Cluster points and accumulation points

A cluster point is a point near which every neighborhood meets a given set or net in a persistent way. Accumulation points are points that can be approached by distinct points of a set. These notions are central to compactness, closedness, and convergence theory.

7 Specialized classes of spaces

Certain spaces have additional structure that makes them especially important in advanced mathematics. These classes often combine topology with other branches of the subject.

7.1 Hausdorff spaces

Hausdorff spaces are spaces in which distinct points can be separated by disjoint neighborhoods. This separation property is strong enough to ensure uniqueness of limits when limits exist. Many standard theorems in topology assume the Hausdorff condition.

7.2 Compact Hausdorff spaces

Compact Hausdorff spaces combine compactness with strong separation. They enjoy many favorable properties, such as well-behaved continuous images and strong forms of function separation. These spaces occupy a central place in both pure topology and functional analysis.

7.3 Metrizable spaces

Metrizable spaces are topological spaces whose topology comes from a metric. They are important because distance makes many arguments concrete and intuitive. A large portion of classical topology can be viewed as the study of which spaces admit such a metric.

7.4 Manifolds

Manifolds are spaces that are locally similar to Euclidean space. They form the natural setting for geometry and many parts of analysis. Their topology can be studied independently of smooth or algebraic structure, though those extra structures are often added later.

7.5 Function spaces

Function spaces consist of functions as points, with topologies chosen to reflect modes of convergence or continuity. They appear in analysis, topology, and mathematical physics. Common examples include spaces of continuous functions and spaces of mappings between topological spaces.

Topology is a broad field with several closely connected branches. These subfields emphasize different structures and methods while sharing common core ideas.

8.1 General topology

General topology studies the abstract properties of topological spaces and continuous maps. It develops the foundational concepts of openness, compactness, connectedness, and separation. This area is also called point-set topology in many contexts.

8.2 Point-set topology

Point-set topology focuses on the detailed behavior of sets of points within topological spaces. It examines bases, closures, continuity, compactness, and convergence in a highly general setting. The subject provides the technical foundation for much of modern topology.

8.3 Algebraic topology

Algebraic topology associates algebraic objects such as groups and rings with topological spaces. These invariants help distinguish spaces that may look similar at first glance. The field studies how global features of a space are reflected in algebraic structure.

8.4 Differential topology

Differential topology studies smooth manifolds and maps between them using topological methods adapted to differentiable settings. It investigates how smooth structure interacts with the underlying topology. The field is closely related to geometry and the study of critical points and embeddings.