General topology

In mathematics, general topology (or point set topology) studies basic ideas used in many parts of topology. It helps us understand other areas like differential topology, geometric topology, and algebraic topology.

The main ideas in point-set topology are continuity, compactness, and connectedness. Continuous functions move nearby points to nearby points. Compact sets are special sets that can be covered by just a few small sets. Connected sets are sets that cannot be split into two far-apart pieces.

These ideas depend on what we mean by "nearby," "small," and "far apart." These meanings are defined using open sets. By changing the definition of an open set, we change what it means for functions to be continuous, sets to be compact, or sets to be connected. Each choice for an open set creates a new kind of structure called a topology. A set with a topology is called a topological space.

One important type of topological space is a metric space. In a metric space, we can measure the distance between any two points. This distance, called a metric, helps make many proofs easier and applies to many common examples of topological spaces.

History

General topology grew from many parts of mathematics. It started from studying the real line. It also came from the idea of a manifold. And it grew from looking at metric spaces, especially those used in functional analysis.

By about 1940, general topology took its present shape. It helps us understand the idea of continuity in many math problems.

A topology on a set

Main article: Topological space

In mathematics, a topology on a set is a way to decide which parts of the set are "open." This helps us understand ideas like closeness and smooth changes. For a set to have a topology, it must include the whole set and the empty set, and it must also allow combining parts in certain ways.

Topologies help us describe spaces in a general way. For example, with real numbers, the usual topology uses small ranges as its basic open sets. This means a set is open if, around every point in it, there’s a tiny range that stays inside the set. Similar ideas work for complex numbers and spaces with more dimensions.

Continuous functions

Main article: Continuous function

In math, a continuous function is one where nearby points stay nearby. Imagine you have a function that moves points from one place to another. If the function is continuous, a small move in the starting place will only cause a small move in the ending place. This helps us understand how things change smoothly without any sudden jumps.

There are different ways to think about continuity. One way uses small areas around points. A function is continuous if, for every small area around a point in the ending place, there’s a matching small area around the starting point that moves inside that ending area. Another way looks at sequences of points. If the function moves the end of sequences to the end of sequences, it’s continuous. These ideas help us study how shapes and spaces behave when we move or change them.

Compact sets

Main article: Compact space

A compact set is a special group of points in space. Think of it like a blanket that can always be covered with a small number of smaller pieces. If you can always find a few pieces from the blanket to cover everything, then the space is compact.

This idea helps mathematicians learn about how spaces work. In simple cases, like a closed line on a number line, the set is compact. This property stays the same even when we change the set smoothly. Compact sets are important in many areas of math.

Connected sets

Main article: connected space

In general topology, we look at how points in a space can be linked together. A space is connected if it cannot be divided into two separate pieces. For example, a line or a circle is connected because you cannot split them into two parts without breaking them.

We also talk about path-connected spaces, where you can draw a continuous path between any two points. All path-connected spaces are connected, but not all connected spaces have paths between every pair of points. For simple shapes like lines or circles, being connected and path-connected mean the same thing.

Products of spaces

Main article: Product topology

When we put many spaces together, we can make a new space using something called the product topology. This helps us see how the parts fit together.

The product topology makes sure that pieces of our new space stay close to each other, just like in the spaces we began with. This idea is helpful in many parts of mathematics.

Separation axioms

Main article: Separation axiom

Separation axioms are rules that help us understand how points in a space can be kept apart. These rules are important in topology, a branch of mathematics that studies spaces and their properties.

In these axioms, we look at how points in a space can be separated by neighborhoods. For example, a Hausdorff space ensures that any two different points can be separated by neighborhoods that don’t overlap. Other axioms, like regular or normal spaces, have similar ideas but with different conditions. These concepts help mathematicians study spaces in a precise way.

Countability axioms

Main article: axiom of countability

Countability axioms are rules that help us understand special types of spaces in mathematics. These rules tell us when we can use countable sets—sets that can be listed, like the numbers 1, 2, 3, and so on—to describe important parts of a space.

Some important countability axioms include:

• Sequential space: a set where certain sequences behave in a special way.
• First-countable space: every point has a countable set of nearby points.
• Second-countable space: the space has a countable basic set of open sets.
• Separable space: there is a countable set that comes close to covering the whole space.
• Lindelöf space: every open covering has a countable smaller covering.
• σ-compact space: the space can be covered by countably many compact sets.

These axioms help us connect different ideas in topology and understand how spaces are built.

Metric spaces

Main article: Metric space

A metric space is a special kind of space that helps us understand distances between points. It has points and a rule to measure the distance between any two points. This rule, called a distance function, must follow a few simple ideas: distances are always zero or more, the distance from a point to itself is zero, the distance from point A to point B is the same as from B to A, and going directly from A to C is never longer than going from A to B and then to C. These ideas help us study shapes and spaces in a clear way.

Baire category theorem

Main article: Baire category theorem

The Baire category theorem is an important idea in mathematics. It helps us understand special kinds of spaces, like complete metric spaces or locally compact Hausdorff spaces. The theorem says that in these spaces, if we put together countably many small sets (nowhere dense sets), their combined inside will still be empty. This means that these special spaces are “big” in a certain way. Also, any open part of a Baire space remains a Baire space itself.

Main areas of research

Continuum theory

Main article: Continuum (topology)

A continuum is a special space that is small and easy to move through. Continuum theory studies these spaces. They appear in many parts of mathematics and help us understand shapes in a general way.

Dynamical systems

Main article: Topological dynamics

Dynamical systems study how spaces and their parts change over time. This helps us understand real-world systems like fluid movement and patterns that repeat in nature.

Pointless topology

Main article: Pointless topology

Pointless topology studies shapes and spaces without focusing on individual points. Instead, it looks at larger regions and their relationships.

Dimension theory

Main article: Dimension theory

Dimension theory studies how we can measure the “dimension” of a space — for example, whether something is one-dimensional like a line, two-dimensional like a flat surface, or three-dimensional like a solid object.

Topological algebras

Main article: Topological algebra

Topological algebras mix ideas from algebra and topology. They look at structures that follow both algebraic and topological rules.

Metrizability theory

Main article: Metrization theorem

Metrizability theory asks whether a space can be described using distances. If a space can be “metrized,” we can use distance rules to understand its structure.

Set-theoretic topology

Main article: Set-theoretic topology

Set-theoretic topology combines ideas from set theory and topology. It looks at questions that depend on the rules of set theory.