Basic Set Theory Question from General Topology by Stephen Willard

I have a desire to study Topology and picked up the book General Topology by Stephan Willard (other recommendations are welcome!) It has an introductory chapter on set theory. I am somewhat familiar with set theory but am stumped on page 6 (this does not bode well for finishing the book :)). I have attached a picture for those who do not have the book. I believe I understand what is a "smallest element" and understand why they are unique if they exist at all. Same for largest element. However, I'm very confused as to what a minimal (maximal) element is, and why you can have a "unique maximal element b which is not a largest element". Part of my problem is I don't understand figure 1.1 at all. Is the vertical line representing real numbers? And what is b in the diagram and why is it connected by a diagonal line? Or it it an arrow showing where b is?

Can someone point me to a another definition of "maximal elmement" or explain it to me and contrast it with "largest element" As always, if there is a better group for what I suspect is a beginner question, please point me to it!

Thanks,

Dave

Solution 1:

In a partial ordering $(A,\le)$, the ordering relation satisfies the following properties for all elements $x,y,z\in A$:

• Reflexivity: $x\le x$
• Antisymmetry: If $x\le y$ and $y\le x$, then $x=y$
• Transitivity: If $x\le y$ and $y\le z$, then $x\le z$

However, the ordering relation need not satisfy the following property:

• Comparability: $x\le y$ or $y\le x$

A partial ordering satisfying this additional property is called a total or linear ordering.

As an example, for any set $X$ the power set $P(X)$ of all subsets of $X$ forms a partial ordering but not in general a total ordering under the inclusion relation $\subseteq$, because if $X$ has at least two elements we can find two subsets where neither one is a subset of the other.

In a partial ordering, to say that an element $x$ is maximal is to say that there is no element greater than it; that is, there is no $y$ with $x<y$ (where "$x<y$" just means $x\le y$ and $x\ne y$). This is weaker than saying that $x$ is largest, which means that it is greater than or equal to every element; that is, $z\le x$ for all $z$. A largest element is maximal, but the converse is not true. In a total ordering, however, the two notions are equivalent.

In the figure, $b$ is maximal because there is nothing greater than it (nothing directly above it), but it is not largest because there are things that it is not greater than.

Solution 2:

The dots represent elements of your partially ordered set. an upward line (vertical or not) between two points represents that the lower point is "smaller" than the upper. So $b$ is the only element that has no element above it, but it is not an element that is above every other.