Basic Set Theory Question from General Topology by Stephen Willard

Question

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

Answer 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.

Answer 2

We're talking here about partial orders. A smallest (largest) element is in fact comparable to all other elements and is smaller (larger) than all of them. A minimal (maximal) element is smaller (larger) than everything that it's comparable to, but it may not be comparable to everything. If follows from this definition that two different minimal (maximal) elements must be incomparable to each other.

An easy example: Consider the partial order of proper subsets of $\Bbb N$ ordered by set inclusion. Then any subset of $\Bbb N$ that contains all but one element is maximal but this partial element has no largest element. If we remove the limitation that the subset must be proper, then $\Bbb N$ itself is the largest element of the proper order.

In the diagram, we're creating a partial order to illustrate a point. The partial order we're creating mostly looks like $\Bbb N$. However, we're adding the element $5a$, which has the property that, like $5, 5a \gt 4$. However, $5a$ is "off to the side" so that it's not comparable to anything above $4$. Then $5a$ is a maximal element because nothing is larger than $5a$, but it's not the largest element because it's not larger than everything (because there are elements that are incomparable to $5a$). Moreover, we've created this example to show that it's possible for a partial order to have exactly one maximal element but no largest element.

Answer 3

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.