Topology

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 \( {\mathbb R}^n, \) 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, \( {\mathbb R}^n,\) there is a standard Euclidean distance function that measures how far apart two points are. This turns \( {\mathbb R}^n\) 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 \( p\) in a metric space is the set of all points within a certain distance of \(p.\)

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 \(X\) be a set. A topology on \(X\) is a collection \( \mathcal T \) of subsets of \(X,\) satisfying the following properties:

1. \(\varnothing\) and \(X\) are both in \( \mathcal T.\)

2. The union of a collection of sets in \(\mathcal T\) is in \(\mathcal T.\)

3. The intersection of finitely many sets in \(\mathcal T\) is in \(\mathcal T.\)

The subsets in \(\mathcal T\) are called open sets.

Every set has two basic (but important) topologies. The discrete topology on \(X\) is the topology \( \mathcal T\) consisting of every subset of \(X\) ("every subset is open"). The indiscrete topology on \(X\) is the topology \(\mathcal T\) such that the only two subsets in \(\mathcal T\) are \( \varnothing\) and \(X\) ("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 \(X\) 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 \(d(x,y)\) that assigns a nonnegative real number distance to any two points \(x,y\) in the space. The conditions that define a metric are

1. \(d(x,y) = 0 \Leftrightarrow x=y.\)
2. \(d(x,y) = d(y,x).\)
3. \(d(x,y)+d(y,z) \ge d(x,z) \) (the triangle inequality: the shortest distance between two points is a straight line).

The topology \(\mathcal T\) induced by a metric \(d\) contains open sets \(U\) with the property that, for any point \(x\in U,\) there is some \( \epsilon>0\) such that all points \(y\) with \(d(x,y) <\epsilon\) are also in \(U.\) That is, any point in \(U\) has a "ball of radius \(\epsilon\)" around it, which is completely contained inside \(U.\)

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 \(x\) and \(y\) can be separated by open sets. That is, there are open sets \(U,V\) with \(x\in U,y\in V, U\cap V = \varnothing.\) To see this, suppose \(d(x,y) = d\); then let \(U\) and \(V\) be the balls of radius \(\frac d2\) around \(x\) and \(y,\) respectively. The fact that \(U\) and \(V\) 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 \( \mathbb R\)) is non-metrizable.

Let \(X\) be a set. A basis for \(X\) is a collection \(\mathcal B\) of subsets of \(X\) such that

1. \( \forall x \in X, \exists B \in \mathcal B : x \in B\)
2. For any two sets \(B_1,B_2 \in \mathcal B\) and point \(x \in B_1 \cap B_2 ,\) there is a third set \(B_3 \in \mathcal B\) such that \(x \in B_3\) and \(B_3\subset B_1 \cap B_2\)

Let \((X,d)\) be a metric space. The collection of balls \(B(x,r)\) where \(x \in X, r > 0\) is a base.

The two properties of a base give the following theorem, whose proof is left as an exercise:

The collection \(\mathcal T\) of sets which are unions of elements in a base \(\mathcal B\) for \(X\) forms a topology on \(X.\) \((\)This is often written "\(\mathcal B\) is a base for \(\mathcal T\)"\(.)\)

The collection of balls \(B(x,r)\) in a metric space is a base for the standard topology coming from the metric.

Let \(\mathcal T\) be the collection of one-point sets of \(X.\) Then \(\mathcal T\) is a base for the discrete topology on \(X.\)

There are several ways to make new topological spaces from old ones. Here are two common ones.

Let \(X\) be a topological space and \(Y\) a subset of \(X.\) The subspace topology on \(Y\) is the topology \( \mathcal T'\) whose open sets are the sets \(U \cap Y,\) where \(U\) is an open set in \(X.\)

It is easy to check that this construction always gives a valid topology on \(Y.\) Note that this introduces some ambiguity about open and closed sets. Saying that a subset of, say, \(\mathbb R\) is open is ambiguous without context: it might be open in some subspace topology but not open inside \(\mathbb R.\)

Find subsets \(Z \subseteq Y\) of \(\mathbb R\) such that \(Z\) is open in \(Y\) with the subspace topology, but \(Z\) is not open in \(\mathbb R.\)

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One trivial example is \(Z=Y=[0,1].\) \(Z\) is not open in \(\mathbb R,\) but since \(Z = {\mathbb R} \cap Y\) and \(\mathbb R\) is open in \(\mathbb R,\) \(Z\) is open in \(Y.\)

The interval \( \left[0,\frac12\right)\) is also open in \(Y,\) since it is the intersection \( \left(-\frac12,\frac12\right) \cap Y.\) \(_\square\)

The second construction involves the Cartesian product of two topological spaces. Recall that the product of two sets \(X\times Y\) consists of ordered pairs \((x,y)\) with \(x\in X,y\in Y.\)

Let \(X,Y\) be topological spaces. Then the product topology on \(X \times Y\) consists of open sets which are unions of sets \( U \times V,\) where \(U \) is open in \(X\) and \(V\) is open in \(Y.\)

So the product topology is the topology whose base consists of subsets \(U \times V,\) with \(U\) open in \(X\) and \(V\) open in \(Y.\)

Note that the open sets are not all of the form \(U \times V.\) It is necessary to take unions, since the union of \( U_1 \times V_1\) and \( U_2 \times V_2\) is not necessarily of the form \(U \times V.\)

Show that the product topology on \({\mathbb R} \times {\mathbb R}\) is the same as the standard Euclidean topology on \( {\mathbb R}^2.\)

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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 \(x\) there is an open rectangle, and inside any open rectangle around \(x\) there is an open disk. So a set that is open in one topology is open in the other. \(_\square\)

The product topology on infinite Cartesian products is more subtle: the product topology on a product \(\prod X_{\alpha}\) has a base consisting of products \( \prod U_\alpha \) where all but finitely many of the \(U_\alpha\) are equal to \(X_{\alpha}.\)

The topology whose base consists of products \(\prod U_{\alpha}\) 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 \(f \colon X \to Y,\) where \(X,Y\) are topological spaces, is continuous if, for every open set \( V\subseteq Y,\) \( f^{-1}(V)\) is open in \(X.\)

This is the correct generalization of the definition of continuity in calculus, which comes from the epsilon-delta definition of a limit.

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 \(f \colon X \to Y\) between topological spaces is called a homeomorphism if \(f\) is continuous and a bijection, and \(f^{-1}\) 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

• simply connected
• normal; Hausdorff; regular, \(T_0,\) etc. (the separation axioms)
• metrizable
• separable, first-countable, second-countable, etc. (the countability conditions).