10.1 Euclidean topology

中文

The archetypal model for a topological space is the real line and the Euclidean plane ${\mathbb{R}}^2$ . On the real line $\mathbb{R}$, an open interval is any set $(a,b)=\{x\in \mathbb{R}|a<x<b\}$ . A set $U\subseteq \mathbb{R}$ is called a neighbourhood of $x\in \mathbb{R}$ if there exists $ϵ>0$ such that the open interval $(x-ϵ,x+ϵ)$ is a subset of $U$. We say a sequence of real numbers $\{x_n\}$ converges to $x\in \mathbb{R}$ , written $x_n\to x$ , if for every $ϵ>0$ there exists an integer $N>0$ such that $|x-x_n|<ϵ$ for all $n>N;$ ; that is, for sufficiently large $n$ the sequence $x_n$ enters and stays in every neighbourhood $U$ of $x$. The point $x$ is then said to be the limit of the sequence $\{x_n\}$

Exercise

Show that the limit of a sequence is unique: if $x_n\to x$ and $x_n\to x^{\prime }$ then $x=x^{\prime }$

Similar definitions apply to the Euclidean plane ${\mathbb{R}}^2$ , where we set $|\mathbf{y}-\mathbf{x}|=\sqrt{(y_1-x_1{)}^2+(y_2-x_2{)}^2}$ . In this case, open intervals are replaced by open ball

$$B_r(\mathbf{x})=\{\mathbf{y}\in {\mathbb{R}}^2||\mathbf{y}-\mathbf{x}|<r\}$$

and a set $U\subseteq {\mathbb{R}}^2$ is said to be a neighbourhood of $\mathbf{x}\in {\mathbb{R}}^2$ if there exists a real number $ϵ>0$ such that the open ball $B_{ϵ}(\mathbf{x})\subseteq U$ . A sequence of points $\left\{{\mathbf{x}}_n\right\}$ converges to $\mathbf{x}\in {\mathbb{R}}^2$ or $x$ is the limit of the sequence $\left\{{\mathbf{x}}_n\right\}$ , if for every $ϵ>0$ there exists an integer $N>0$ such that

$${\mathbf{x}}_n\in B_{ϵ}(\mathbf{x})\text{ for all }n>N.$$

Again we write ${\mathbf{x}}_n\to \mathbf{x}$ and the definition is equivalent to the statement that for every neighbourhood $U$ of $x$ there exists $N>0$ such that ${\mathbf{x}}_n\in U$ for all $n>N$

An open set $U$ in $\mathbb{R}$ or ${\mathbb{R}}^2$ is a set that is a neighbourhood of every point in it. Intuitively, $U$ is open in $\mathbb{R}$ (resp. ${\mathbb{R}}^2)$ if every point in $U$ can be ‘thickened out’ to an open interval (resp. open ball) within $U$ (see Fig. 10.1). For example, the unit ball $B_1(\mathbf{O})=\{\mathbf{y}||\mathbf{y}{|}^2<1\}$ is an open set since, for every point $\mathbf{x}\in B_1(\mathbf{O})$ the open ball $B_{ϵ}(\mathbf{x})\subseteq B_1(\mathbf{O})$ where $ϵ=1-|\mathbf{x}|>0$

Figure 10.1 Points in an open set can be ‘thickened’ to an open ball within the set

On the real line it may be shown that the most general open set consists of a union of non-intersecting open intervals,

$$\dots ,(a_{-1},a_0),(a_1,a_2),(a_3,a_4),(a_5,a_6),\dots$$

where $\begin{array}{r}\dots a_{-1}<a_0\le a_1<a_2\le a_3<a_4\le a_5<a_6\le ..\end{array}$ . In ${\mathbb{R}}^2$ open sets cannot be so simply categorized, for while every open set is a union of open balls, the union need not be disjoint.

In standard analysis, a function $f:\mathbb{R}\to \mathbb{R}$ is said to be continuous at $x$ if for every $ϵ>0$ there exists $\delta >0$ such that

$$|y-x|<\delta ⟹|f(y)-f(x)|<ϵ.$$

Hence, for every $ϵ>0$ , the inverse image set $f^{-1}(f(x)-ϵ,f(x)+ϵ)$ is a neighbourhood of $x$, since it includes an open interval $(x-\delta ,x+\delta )$ ) centred on $x$. As every neighbourhood of $f(x)$ contains an interval of the form $(f(x)-ϵ,f(x)+ϵ)$ the function $f$ is continuous at $x$ if and only ifthe inverse image of every neighbourhood of $f(x)$ is a neighbourhood of $x$. A function $f:\mathbb{R}\to \mathbb{R}$ is said to be continuous on $\mathbb{R}$ if it is continuous at every point $x\in \mathbb{R}$

Theorem 10.1 · Continuity characterized by inverse images of open sets

A function $f:\mathbb{R}\to \mathbb{R}$ is continuous on $\mathbb{R}$ if and only ifthe inverse image $V=f^{-1}(U)$ of every open set $U\subseteq \mathbb{R}$ is an open subset of $R$

Proof

Let $f$ be continuous on $\mathbb{R}$. Since an open set $U$ is a neighbourhood of every point $y\in U,$ , its inverse image $V=f^{-1}(U)$ must be a neighbourhood of every point $x\in V$ Hence $V$ is an open set.

Conversely let $f:\mathbb{R}\to \mathbb{R}$ be any function having the property that $V=f^{-1}(U)$ is open for every open set $U\subseteq \mathbb{R}$ . Then for any $x\in \mathbb{R}$ and every $ϵ>0$ the inverse image under $f$ of the open interval $(f(x)-ϵ,f(x)+ϵ)$ is an open set including $x$. It therefore contains an open interval of the form $(x-\delta ,x+\delta )$ ), so that $f$ is continuous at $x$. Since $x$ is an arbitrary point, the function $f$ is continuous on $\mathbb{R}$ .

In general topology this will be used as the defining characteristic of a continuous map. In ${\mathbb{R}}^2$ the treatment is almost identical. A function $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is said to be continuous at $x$ if for every $ϵ>0$ there exists a real number $\delta >0$ such that

$$|\mathbf{y}-\mathbf{x}|<\delta ⟹|f(\mathbf{y})-f(\mathbf{x})|<ϵ.$$

An essentially identical proof to that given in Theorem 10.1 shows that a function $f$ is continuous on ${\mathbb{R}}^2$ if and only if the inverse image $f^{-1}(U)$ of every open set $U\subseteq {\mathbb{R}}^2$ is an open subset of ${\mathbb{R}}^2$ . The same applies to real-valued functions $f:{\mathbb{R}}^2\to \mathbb{R}$ . Thus continuity of functions can be described entirely by their inverse action on open sets. For this reason, open sets are regarded as the key ingredients of a topological space. Experience from Euclidean spaces and surfaces embedded in them has taught mathematicians that the most important properties of open sets can be summarized in a few simple rules, which are set out in the next section (see also [1–8]).