2.3 Topological spaces

The most general structure with which we work is a topological space. Physicists often tend to think that all the spaces they deal with are equipped with metrics. However, this is not always the case. In fact, metric spaces form a subset of manifolds and manifolds form a subset of topological spaces.

2.3.1 Definitions

Definition 2.3

Let $X$ be any set and $\mathcal{T}=\{U_i|i\in I\}$ denote a certain collection of subsets of $X$ . The pair $(X,\mathcal{T})$ is a topological space if $\mathcal{T}$ satisfies the following requirements.

(i) $\varnothing ,X\in \mathcal{T}$ .
(ii) If $\mathcal{T}$ is any (maybe infinite) subcollection of $I$ , the family $\{U_j|j\in J\}$ satisfies ${\cup }_{j\in J}{U}_j\in \mathcal{T}$ .
(iii) If $K$ is any finite subcollection of $I$ , the family $\{U_k|k\in K\}$ satisfies ${\cap }_{k\in K}{U}_k\in \mathcal{T}$ .

$X$ alone is sometimes called a topological space. The $U_i$ are called the open sets and $\mathcal{T}$ is said to give a topology to $X$ .

Example 2.7

(a) If $X$ is a set and $\mathcal{T}$ is the collection of all the subsets of $X$ , then (i)-(iii) are automatically satisfied. This topology is called the discrete topology.

(b) Let $X$ be a set and $\mathcal{T}=\{\varnothing ,X\}$ . Clearly $\mathcal{T}$ satisfies (i)-(iii). This topology is called the trivial topology. In general the discrete topology is too stringent while the trivial topology is too trivial to give any interesting structures on $X$ .

(c) Let $X$ be the real line $\mathbb{R}$ . All open intervals $(a,b)$ and their unions define a topology called the usual topology; $a$ and $b$ may be $-\infty$ and $\infty$ respectively. Similarly, the usual topology in ${\mathbb{R}}^n$ can be defined. [Take a product $(a_1,b_1)\times \cdots \times (a_n,b_n)$ and their unions…]

Exercise 2.13

In definition 2.3, axioms (ii) and (iii) look somewhat unbalanced. Show that, if we allow infinite intersection in (iii), the usual topology in $\mathbb{R}$ reduces to the discrete topology (and is thus not very interesting).

A metric $d:X\times X\to \mathbb{R}$ is a function that satisfies the conditions:

(i) $d(x,y)=d(y,x)$
(ii) $d(x,y)\ge 0$ where the equality holds if and only if $x=y$
(iii) $d(x,y)+d(y,z)\ge d(x,z)$

for any $x,y,z\in X$ . If $X$ is endowed with a metric $d$ , $X$ is made into a topological space whose open sets are given by ‘open discs’,

$$U_{\epsilon }(X)=\{y\in X|d(x,y)<\epsilon \}\text{\hspace{1em}(2.25)}$$

and all their possible unions. The topology $\mathcal{T}$ thus defined is called the metric topology determined by $d$ . The topological space $(X,\mathcal{T})$ is called a metric space. [Exercise: Verify that a metric space $(X,\mathcal{T})$ is indeed a topological space.]

Let $(X,\mathcal{T})$ be a topological space and $A$ be any subset of $X$ . Then $\mathcal{T}=\{U_i\}$ induces the relative topology in $A$ by ${\mathcal{T}}^{\prime }=\{U_i\cap A|U_i\in \mathcal{T}\}$ .

Example 2.8

Let $X={\mathbb{R}}^{n+1}$ and take the $n$ -sphere $S^n$ ,

$$(x^0{)}^2+(x^1{)}^2+\cdots +(x^n{)}^2=1.\text{\hspace{1em}(2.26)}$$

A topology in $S^n$ may be given by the relative topology induced by the usual topology on ${\mathbb{R}}^{n+1}$ .

2.3.2 Continuous maps

Definition 2.4

Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ is continuous if the inverse image of an open set in $Y$ is an open set in $X$ .

This definition is in agreement with our intuitive notion of continuity. For instance, let $f:\mathbb{R}\to \mathbb{R}$ be defined by

$$f(x)=\{\begin{array}{ll}-x+1& x\le 0\\ -x+\frac{1}{2}& x>0.\end{array}\text{\hspace{1em}(2.27)}$$

We take the usual topology in $\mathbb{R}$ , hence any open interval $(a,b)$ is an open set. In the usual calculus, $f$ is said to have a discontinuity at $x=0$ . For an open set $(3/2,2)\subset Y$ , we find $f^{-1}((3/2,2))=(-1,-1/2)$ which is an open set in $X$ . If we take an open set $(1-1/4,1+1/4)\subset Y$ , however, we find $f^{-1}((1-1/4,1+1/4))=(-1/4,0]$ which is not an open set in the usual topology.

Exercise 2.14

By taking a continuous function $f:\mathbb{R}\to \mathbb{R}$ , $f(x)=x^2$ as an example, show that the reverse definition, ‘a map $f$ is continuous if it maps an open set in $X$ to an open set in $Y$ ’, does not work. [Hint: Find where $(-\epsilon ,+\epsilon )$ is mapped to under $f$ .]

2.3.3 Neighbourhoods and Hausdorff spaces

Definition 2.5

Suppose $\mathcal{T}$ gives a topology to $X$ . $N$ is a neighbourhood of a point $x\in X$ if $N$ is a subset of $X$ and $N$ contains some (at least one) open set $U_i$ to which $x$ belongs. (The subset $N$ need not be an open set. If $N$ happens to be an open set in $\mathcal{T}$ , it is called an open neighbourhood.)

Example 2.9

Take $X=\mathbb{R}$ with the usual topology. The interval $[-1,1]$ is a neighbourhood of an arbitrary point $x\in (-1,1)$ .

Definition 2.6

A topological space $(X,\mathcal{T})$ is a Hausdorff space if, for an arbitrary pair of distinct points $x,x^{\prime }\in X$ , there always exist neighbourhoods $U_x$ of $x$ and $U_{x^{\prime }}$ of $x^{\prime }$ such that $U_x\cap U_{x^{\prime }}=\varnothing$ .

Exercise 2.15

Let $X=\{\text{John, Paul, Ringo, George}\}$ and $U_0=\varnothing$ , $U_1=\{\text{John}\},U_2=\{\text{John, Paul}\},U_3=\{\text{John, Paul, Ringo, George}\}$ . Show that $\mathcal{T}=\{U_0,U_1,U_2,U_3\}$ gives a topology to $X$ . Show also that $(X,\mathcal{T})$ is not a Hausdorff space.

Unlike this exercise, most spaces that appear in physics satisfy the Hausdorff property. In the rest of the present book we always assume this is the case.

Exercise 2.16

Show that $\mathbb{R}$ with the usual topology is a Hausdorff space. Show also that any metric space is a Hausdorff space.

2.3.4 Closed set

Let $(X,\mathcal{T})$ be a topological space. A subset $A$ of $X$ is closed if its complement in $X$ is an open set, that is $X-A\in \mathcal{T}$ . According to the definition, $X$ and $\varnothing$ are both open and closed. Consider a set $A$ (either open or closed). The closure of $A$ is the smallest closed set that contains $A$ and is denoted by $\bar{A}$ . The interior of $A$ is the largest open subset of $A$ and is denoted by $A^{\circ }$ . The boundary $b(A)$ of $A$ is the complement of $A^{\circ }$ in $A$ ; $b(A)=A-A^{\circ }$ . An open set is always disjoint from its boundary while a closed set always contains its boundary.

Example 2.10

Take $X=\mathbb{R}$ with the usual topology and take a pair of open intervals $(-\infty ,a)$ and $(b,\infty )$ where $a<b$ . Since $(-\infty ,a)\cup (b,\infty )$ is open under the usual topology, the complement $[a,b]$ is closed. Any closed interval is a closed set under the usual topology. Let $A=(a,b)$ , then $\bar{A}=[a,b]$ . The boundary $b(A)$ consists of two points $\{a,b\}$ . The sets $(a,b),[a,b],(a,b]$ , and $[a,b)$ all have the same boundary, closure and interior. In ${\mathbb{R}}^n$ , the product $[a_1,b_1]\times \cdots \times [a_n,b_n]$ is a closed set under the usual topology.

Exercise 2.17

Whether a set $A\subset X$ is open or closed depends on $X$ . Let us take an interval $I=(0,1)$ in the $x$ -axis. Show that $I$ is open in the $x$ -axis $\mathbb{R}$ while it is neither closed nor open in the $xy$ -plane ${\mathbb{R}}^2$ .

2.3.5 Compactness

Let $(X,\mathcal{T})$ be a topological space. A family $\{A_i\}$ of subsets of $X$ is called a covering of $X$ , if

$$\bigcup _{i\in I}{A}_i=X.$$

If all the $A_i$ happen to be the open sets of the topology $\mathcal{T}$ , the covering is called an open covering.

Definition 2.7

Consider a set $X$ and all possible coverings of $X$ . The set $X$ is compact if, for every open covering $\{U_i|i\in I\}$ , there exists a finite subset $J$ of $I$ such that $\{U_j|j\in J\}$ is also a covering of $X$ .

In general, if a set is compact in ${\mathbb{R}}^n$ , it must be bounded. What else is needed? We state the result without the proof.

Theorem 2.3

Let $X$ be a subset of ${\mathbb{R}}^n$ . $X$ is compact if and only if it is closed and bounded.

Example 2.11

(a) A point is compact.

(b) Take an open interval $(a,b)$ in $\mathbb{R}$ and choose an open covering $U_n=(a,b-1/n)$ , $n\in \mathbb{N}$ . Evidently

$$\bigcup _{n\in \mathbb{Z}}{U}_n=(a,b).$$

However, no finite subfamily of $\{U_n\}$ covers $(a,b)$ . Thus, an open interval $(a,b)$ is non-compact in conformity with theorem 2.3.

(c) $S^n$ in example 2.8 with the relative topology is compact, since it is closed and bounded in ${\mathbb{R}}^{n+1}$ .

The reader might not appreciate the significance of compactness from the definition and the few examples given here. It should be noted, however, that some mathematical analyses as well as physics become rather simple on a compact space. For example, let us consider a system of electrons in a solid. If the solid is non-compact with infinite volume, we have to deal with quantum statistical mechanics in an infinite volume. It is known that this is mathematically quite complicated and requires knowledge of the advanced theory of Hilbert spaces. What we usually do is to confine the system in a finite volume $V$ surrounded by hard walls so that the electron wavefunction vanishes at the walls, or to impose periodic boundary conditions on the walls, which amounts to putting the system in a torus, see example 2.5(b). In any case, the system is now put in a compact space. Then we may construct the Fock space whose excitations are labelled by discrete indices. Another significance of compactness in physics will be found when we study extended objects such as instantons and Belavin-Polyakov monopoles, see section 4.8. In field theories, we usually assume that the field approaches some asymptotic form corresponding to the vacuum (or one of the vacua) at spatial infinities. Similarly, a class of order parameter distributions in which the spatial infinities have a common order parameter is an interesting class to study from various points of view as we shall see later. Since all points at infinity are mapped to a point, we have effectively compactified the non-compact space ${\mathbb{R}}^n$ to a compact space $S^n={\mathbb{R}}^n\cup \{\infty \}$ . This procedure is called the one-point compactification.

2.3.6 Connectedness

Definition 2.8

(a) A topological space $X$ is connected if it cannot be written as $X=X_1\cup X_2$ , where $X_1$ and $X_2$ are both open and $X_1\cap X_2=\varnothing$ . Otherwise $X$ is called disconnected.

(b) A topological space $X$ is called arcwise connected if, for any points $x,y\in X$ , there exists a continuous map $f:[0,1]\to X$ such that $f(0)=x$ and $f(1)=y$ . With a few pathological exceptions, arcwise connectedness is practically equivalent to connectedness.
(c) A loop in a topological space $X$ is a continuous map $f:[0,1]\to X$ such that $f(0)=f(1)$ . If any loop in $X$ can be continuously shrunk to a point, $X$ is called simply connected.

Example 2.12

(a) The real line $\mathbb{R}$ is arcwise connected while $\mathbb{R}-\{0\}$ is not. ${\mathbb{R}}^n$ $(n\ge 2)$ is arcwise connected and so is ${\mathbb{R}}^n-\{0\}$ .

(b) $S^n$ is arcwise connected. The circle $S^1$ is not simply connected. If $n\ge 2$ , $S^n$ is simply connected. The $n$ -dimensional torus

$$T^n=\underset{n}{\underbrace{S^1\times S^1\times \cdots \times S^1}}(n\ge 2)$$

is arcwise connected but not simply connected.

(c) ${\mathbb{R}}^2-\mathbb{R}$ is not arcwise connected. ${\mathbb{R}}^2-\{0\}$ is arcwise connected but not simply connected. ${\mathbb{R}}^3-\{0\}$ is arcwise connected and simply connected.

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