Real Analysis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 19 July 2014

Lecture 29

Topology

Goal: Understand and prove the Intermediate value theorem and The max-min theorem.

(Intermediate Value Theorem) Let $f:[a,b]\to ℝ$ be a continuous function. If $x\in ℝ$ and $x$ is between $f\left(a\right)$ and $f\left(b\right)$ then there exists $c\in [a,b]$ with $f\left(c\right)=x\text{.}$

(Max-Min theorem) Let $f:[a,b]\to ℝ$ be a continuous function. Then there exists $m,M\in ℝ$ such that $$f\left([a,b]\right)=[m,M]\text{.}$$

Let $X$ be a set.

A topology on $X$ is a collection $𝒯$ of subsets of $X$ such that

(a)	$\varnothing \in 𝒯$ and $X\in 𝒯,$
(b)	if $𝒮\subseteq 𝒯$ then $\bigcup _{u\in 𝒮}u\in 𝒯,$
(c)	if $1,\dots ,n\in {\mathbb{Z}}_{>0}$ and $U_1,\dots ,U_n\in 𝒯$ then $U_1\cap \cdots \cap U_n\in 𝒯\text{.}$

A topological space is a set $X$ with a topology on $X\text{.}$

Let $X$ be a topological space with topology $𝒯\text{.}$ Let $E$ be a subset of $X\text{.}$

The set $E$ is open is $E\in 𝒯\text{.}$

Let $X=\{a,b,c,d\}\text{.}$ $$\begin{array}{ccc}{\displaystyle 𝒯}& =& {\displaystyle \left\{\genfrac{}{}{0ex}{}{\varnothing ,\left\{a\right\},\{b,c\},\{a,b,c\},}{\left\{d\right\},\{b,c,d\},\{a,b,c,d\}}\right\}\text{is not a topology on}\hspace{0.17em}X\text{.}}\\ {\displaystyle 𝒯}& =& {\displaystyle \left\{\genfrac{}{}{0ex}{}{\varnothing ,\left\{a\right\},\left\{d\right\},\{b,c\},\{a,d\},}{\{a,b,c\},\{b,c,d\},\{a,b,c,d\}}\right\}\text{is a topology on}\hspace{0.17em}X\text{.}}\end{array}$$

Let $X={ℝ}^n\text{.}$ $$𝒯=\{\text{open subsets}\hspace{0.17em}E\subseteq {ℝ}^n\}$$ is a topology on ${ℝ}^n$ (a subset $E\subseteq {ℝ}^n$ is open if $E$ is a union of $\epsilon \text{-balls).}$

Let $X$ be a topological space with topology $𝒯\text{.}$

Let $E$ be a subset of $X\text{.}$

The set $E$ is closed if $E^c$ is open.

Recall: $E^c=\{x\in X\hspace{0.17em}|\hspace{0.17em}x\notin E\}\text{.}$

The set $E$ is connected if there do not exist open sets $U_1,U_2$ such that $$U_1\cup U_2\supseteq E\text{and}(U_1\cap E)\cap (U_2\cap E)=\varnothing \text{.}$$

The set $E$ is compact is $E$ satisfies: If $𝒮\subseteq 𝒯$ and $\left(\bigcup _{u\in 𝒮}u\right)\supseteq E$ then there exists $n\in {\mathbb{Z}}_{>0}$ and $U_1,\dots ,U_n\in 𝒮$ such that $(U_1\cup U_2\cup \cdots \cup U_n)\supseteq E\text{.}$ In English: $E$ is compact if every open cover of $E$ has a finite subcover.

What does this have to do with the Max-min theorem?

Let $X=ℝ$ with the standard topology. Let $E\subseteq ℝ\text{.}$ Then $E$ is compact and connected
if and only if
there exist $m,M\in ℝ$ such that $E=[m,M]\text{.}$

Continuous functions for topological spaces

Let $X$ and $Y$ be sets. Let $f:X\to Y$ be a function. Let $E$ be a subset of $X\text{.}$ The image of $E$ is $$f\left(E\right)=\left\{f\left(x\right)\hspace{0.17em}\right|\hspace{0.17em}x\in E\}\text{.}$$ $$\begin{array}{c} X Y E f(E) \end{array}$$ Let $F$ be a subset of $Y\text{.}$ the inverse image of $F$ is $$f^{-1}\left(F\right)=\{x\in X\hspace{0.17em}|\hspace{0.17em}f\left(x\right)\in F\}\text{.}$$ $$\begin{array}{c} X Y F f-1(F) \end{array}$$

Warning: Despite the confusing notation, $f^{-1}\left(F\right)$ has nothing to do with the inverse function to $f\text{.}$

Let $X$ be a topological space with topology $𝒯\text{.}$ Let $Y$ be a topological space with topology $𝒵\text{.}$ Let $f:X\to Y$ be a function. The function $f:X\to Y$ is continuous if $f$ satisfies: If $V\in 𝒵$ then $f^{-1}\left(V\right)\in 𝒯\text{.}$ In English: Inverse images of open sets are open.

Let $a,b\in ℝ$ with $a<b\text{.}$ Let $X=[a,b]$ with the topology given by $E\subseteq X$ is open if $E$ is a union of $\epsilon \text{-balls.}$ Let $Y=ℝ$ with the standard topology.

Let $f:[a,b]\to ℝ\text{.}$ The function $f$ is continuous (as a function between topological spaces) if and only if $f$ satisfies if $c\in [a,b]$ then $\underset{x\to c}{\text{lim}}f\left(x\right)=f\left(c\right)\text{.}$

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100517Lect29.pdf and was given on 17 May 2010.

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