Extension: Heine-Borel Proof of Extreme Value Theorem

Definition: Compact sets, connected sets, closed sets

Definition 1a: A set $X$ is compact iff every open covering of $X$ has a finite sub-covering. In the unification of EVT gives a more detailed definition of open covering and finite sub-covering.

Definition 1b: We define connected sets by defining its dual concept dis-connected: A set $Y$ is dis-connected iff we can split $Y$ into two open, disjoint, non-empty sets $U,V$. And if a set $X$ is not dis-connected we say it is connected.

Definition 1c: The previous dual concept definition works for closed: A set $Y$ is open in $A$ (where $A$ is the whole set and may be omitted when it is clear) iff for every $y\in Y$, there exists a $\epsilon >0$ such that all points in $A$ with distance (which can be thought as normal distance of two points in $\mathbb{R}$) less than $\epsilon$ is in $Y$. And a set $X$ is closed in $A$ iff the set $A\backslash X$ is open in $A$.

Theorem (Heine-Borel in $\mathbb{R}$): A set in $\mathbb{R}$ is compact if and only if it is closed and bounded.

Example 1: Any closed interval $[a,b]$ is closed and bounded, so compact.

Example 2: $\mathbb{R}$ is closed, but is not bounded, so is not compact.

Example 3: The open interval $(a,b)$ is bounded but not closed, so not compact. The intervals $(a,b]$ and $[a,b)$ are not closed (and not open), so are not compact.

Example 4: Any finite union of closed bounded intervals $[a,b]$ is closed, so is compact.

Proposition 1a: Continuous functions map compact sets to compact sets

Proposition 1b: Continuous functions map connected sets to connected sets.

TODO explain prop 1

Proposition 2: The connected sets in $\mathbb{R}$ are exactly the intervals.

TODO explain propositions

Theorem (Extreme Value Theorem): A continuous function $f:[a,b]\to\mathbb{R}$ attains its maximum and minimum.

Proof: $f$ is continuous and $[a,b]$ is both compact and connected, so by Proposition 1, its image $f(X)=\{y\in\mathbb{R}:f(x)=y\text{ for some }x\in[a,b]\}$ is also compact and connected. $f(X)$ connected means it is an interval (Prop. 2), and $f(X)$ compact means it is closed and bounded (Heine-Borel Theorem). Thus $f(X)$ is a closed, bounded interval, say $f(X)=[c,d]$, so $f$ attains its maximum $d$ and minimum $c$.

The Extreme Value Theorem can be generalized to spaces other than $\mathbb{R}$. This generalization is usually covered in point-set topology.

Theorem (generalized Extreme Value Theorem): A continuous function $f:X\to Y$, for $X$ compact, attains its maximum and minimum.

Here $X$ and $Y$ are objects called topological spaces, with the requirement that $Y$ also has something called the order topology. Intuitively, topological spaces are spaces where we can talk about open and closed sets. The real line $\mathbb{R}$ is an example of a topological space. At least the generalized Extreme Value Theorem looks pretty similar to the case for $\mathbb{R}$, right? The proof is a bit different though (can't use the Heine-Borel theorem for the more general topological space $X$!).