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Introduction to Topology, Part 1: Set Theory.


Motivation.

Topology, to me, has a flexible, somewhat-vague meaning; this isn't unreasonable, considering what topology is. For me, topology generalizes the idea of closeness: if two things are close to each other, in some (perhaps non-physical, non-intuitive) sense, we group them together. Elements which are grouped together more often are considered somehow closer together than ones which are not grouped together so often.

To others (and as it is generally introduced in textbooks), topology essentially studies continuity of maps and connectedness of structures. For continuity of maps: there's a huge number of fields of mathematics which require continuity of maps, and, for each, continuity means something slightly different — topology standardizes these definitions and attempts to derive general information about continuity. For connectedness of structures: in geometry, shapes are rigid structures which have definite lengths, areas, and so forth; on the other hand, for topology, every shape is thought of to be made of rubber or clay: you can mold and reshape it so long as you don't tear it or poke holes in it. For example, as we will see, the square and the circle are the "same" shape topologically (just round out the edges of the square and reshape it into a circle). This might seem a bit strange: what could we gain from looking at shapes in such a way? The vague answer is: how connected (or "how many holes") a structure has. Neat.

Note that this introduction is just a kind of vague, hand-wavy introduction: there's a ton of information in the field of topology, and a number of different tools that we can build up. One of the most exciting things about topology is that it can tell us about the world we live in or the universe around us just based on the general shape we think these things may be—

Set Theory in a Nutshell.

We need to get some definitions out of the way before we start doing topology. I won't exhaust the subject, so there may be things we need to note later. For now, let's just do some basic definitions and basic properties.

Definition (Set). A set $X$ is taken to be a collection of elements; we denote it by putting its elements between braces like this: $X = \{3,4,5,6,...\}$. We may put anything into sets (even other sets!) but the only real restriction is that each element can only be in a set once. A subset $A$ of some set $X$ is itself a set consisting of some or all elements of $X$; we write $A\subseteq X$ in this case.

For example, the set of positive whole numbers less than 4 is equal to $\{1,2,3\}$. We can look at the subsets of this set, too. All the one-element subsets are: $\{1\}, \{2\}, \{3\}$. All the two element subsets are: $\{1,2\}, \{1,3\}, \{2,3\}$. All the three element subsets are: $\{1,2,3\}$. We also have a "special" subset, which is a subset of every set which we call "the empty set" and denote by $\emptyset$. This is considered a 0-element set most of the time.

Notationally, we can say that \[1\in \{1,2,3\}\] ("1 is an element of $\{1,2,3\}$" or "1 is in $\{1,2,3\}$") and \[\{1,2\}\subseteq \{1,2,3\}\] ("$\{1,2\}$ is a subset of $\{1,2,3\}$"). Note, though, that we cannot say $\{1,2\}\in \{1,2,3\}$ because this says the set $\{1,2\}$ is an element of the set $\{1,2,3\}$. On the other hand, if we had the set $\{1,2,3,\{a,b\}$ which contained the numbers $1, 2$, and $3$ as well as the set $\{a,b\}$ we could say \[\{a,b\}\in \{1,2,3,\{a,b\}\}\] since $\{a,b\}$ (even though it is, itself, a set) is an element of the set $\{1,2,3,\{a,b\}\}$.
 
Can we say that $\{1,2\}\in\{1,2,3,\{1,2\}\}$? Yes we can — why?
 
Can we say that $\{1,2,3\}\in\{1,2,3,\{1,2\}\}$? No — why not?
 
Can we say that $\{1,2,3\}\subseteq \{1,2,3,\{1,2\}\}$? Yep — why?
 
Can we say that $\{1,2,3\}\subseteq \{1,2,\{1,2,3\}\}$? Nope — why not?



We also have operations we can do on sets. There are many, but we'll stick to just four. First, we'll draw our sets as circles which intersect (a Venn Diagram) to try to give an example of each of these operations.


Definition (Union). Given two sets, $A$ and $B$, the union of the sets is every element which is in $A$ or $B$ or both; it is denoted by $A\cup B$.

For example, the union of the circles above is everything part of at least one circle: the red part, the blue part, and the purple part in the middle. For another example, given the sets $\{1,2,3\}$ and $\{3,4,5\}$ the union is \[\{1,2,3\}\cup \{3,4,5\} = \{1,2,3,4,5\}.\]

Definition (Intersection). Given two sets, $A$ and $B$, the intersection of the sets is every element which is in both $A$ and $B$; it is denoted by $A\cap B$.

For example, the intersection of the circles above is only the purple part in the center. For another example, given the sets $\{1,2,3\}$ and $\{3,4,5\}$ the intersection is \[\{1,2,3\}\cap \{3,4,5\} = \{3\}.\]

Definition (Complement). Given a set $A$, the complement of $A$ is every element which is not in $A$; it is denoted $A^{C}$.

Note that, in order to define the complement, we need a universe of elements for $A$ to be in. If we defined it to be literally everything not in $A$ then it wouldn't be so useful of a concept. For example, if we said, "The set $A = \{0,1,2,3\}$ is a subset of the integers," and asked for $A^{C}$, the solution would be $\{\dots, -2, -1, 4, 5, 6, 7, \dots\}$ since these are all of the integers not in $A$. On the other hand, if we said, "The set $A=\{0,1,2,3\}$ is a subset of the real numbers," then the complement would be every single real number except for $0,1,2,3$; in particular, $\pi, e, \sqrt{2}\in A^{C}$. Thus, the complement of $A$ depends on the space containing $A$.

For example, the complement of the circle $A$ above is only the pure blue part on the right side, not including the middle purple part. This consists of every element not in the circle $A$.



Before moving on, let's just introduce one last topic: one particularly nice way of "building" or "defining" sets.

Definition (Set-Builder Notation). We may define a set $A$ using the following notation, \[A = \{x\in U\,|\, P(x)\}\] where the first half between the brackets means, "$x$ is an element of the universe $U$" (usually taken to be whatever set $A$ is a subset of), and the second half after the vertical bar is some property that $x$ must satisfy. See the examples below to clear this up.

It's much easier to give examples of this than to define it. For example, \[A = \{x\in {\mathbb N}\,|\, x \lt 5\}\] is said, "$A$ is the set consisting of the $x$'s in the natural numbers with the property that $x \lt 5$." In this case, $A = \{1,2,3,4\}$ because these are the only natural numbers less than 5.

[Note: I take ${\mathbb N} = \{1,2,3,4,5,\dots\}$ to be the set of natural numbers, but some sources will include $0$ in the set of natural numbers. There are good arguments for and against including $0$, but I prefer it without $0$.]

The universe is important in the set-builder notation, since taking nearly the same set as above with a different universe, \[B = \{x\in {\mathbb Z}\,|\, x \lt 5\}\] where \[{\mathbb Z} = \{\dots, -2,-1,0,1,2,\dots\}\] is standard shorthand for the set of all integers, gives us a different set altogether: \[B = \{\dots, -3,-2,-1,0,1,2,3,4\}\] so $B$ contains each negative whole number and $0$ in addition to the elements that were in $A$. If we were to say, \[C = \{x\in {\mathbb R}\,|\, x\lt 5\}\] this would have even more elements than $B$ does! What are some of the elements in $C$ that aren't in $B$ or $A$? Here's some: $\pi, -2\pi, -e^{\pi}, \sqrt{2}, \dots$

We can have more complicated sets as well. For example, \[D = \{x\in {\mathbb N}\,|\, x\mbox{ is a prime number.}\}\] is the set $D = \{2,3,5,7,11,\dots\}$. Or, we could combine properties, as in \[E = \{x\in {\mathbb N}\,|\, x > 1 \mbox{ and } x \leq 5\}\] which is the set $E = \{2,3,4,5\}$. How would this be different if, when defining $E$, instead of $x\in {\mathbb N}$ we asked for $x\in {\mathbb R}$?

Properties and Proofs.

In general, there's a few common techniques one can use to prove statements in set theory. Let's just go through some simple ones. For all of these, $A,B$ are sets and $x,y$ are elements. Usually it is the case that we use capital letters for sets and lower-case letters for elements.

To prove that $x\in A$...One is usually given a property for $A$, so one needs only to show that $x$ satisfies that property. For example, if $A = \{x\in {\mathbb Z}\,|\, x \mbox{ is even.}\}$ then $2\in A$ because 2 is even, but $3\notin A$ (this is the symbol for "not in") because 3 is not even.

To prove that $x\in A\cap B$... One shows that $x$ is in both $A$ and $B$.

To prove that $x\in A\cup B$... One shows that $x$ is either in $A$, or $B$, or perhaps both.

To prove that $x\in A^{C}$... One shows that $x$ is not in $A$.

To prove that $B\subseteq A$... This one is a bit harder; this says that $B$ is a subset of $A$ or, in other words, every element of $B$ is an element of $A$. Using this second part, we have the following method: take some arbitrary element $y\in B$ and show that $y$ is also in $A$. Sometimes, you may need to do this for each element of $B$.

To prove that $A = B$... One shows that $A\subseteq B$ and $B\subseteq A$. Make sure you think about this one before moving on: $A$ is equal to $B$ if and only if every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

These last two methods are so common that I'm going to take some time to do some examples.

Example. \[\{n\in {\mathbb N}\,|\, n\mbox{ is odd}.\} = \{n\in {\mathbb N}\,|\, n^{2}\mbox{ is odd.}\}.\]

We need to show that each set is a subset of the other. We take an arbitrary element in the left-hand set and show it's in the right-hand set: so, take $x\in \{n\in {\mathbb N}\,|\, n\mbox{ is odd}.\}$. Since $x$ is odd, we can either note that $\mbox{odd } \times \mbox{ odd } = \mbox{ odd}$, or we can prove it explicitly; let's do the latter. If $x$ is odd, then $x = 2k + 1$ for some $k\in {\mathbb Z}$; then we note \[x^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^{2} + 2k) + 1\] which shows that $x^2$ is odd. Hence, $x\in \{n\in {\mathbb N}\,|\, n^{2}\mbox{ is odd.}\}$. This implies the left-hand set is a subset of the right-hand set; we now need to show the reverse containment.
 
Suppose now that $x\in \{n\in {\mathbb N}\,|\, n^{2}\mbox{ is odd.}\}$, which implies that $x^2$ is odd. We need to show this is in the left-hand set, so we need to show that $x$ is odd. Suppose that, instead, $x$ is even: then $x = 2k$ for some $k\in {\mathbb Z}$, which implies $x^2 = 4k^2 = 2(2k^{2})$ which is even — but this is a contradiction, since we had assumed that $x^2$ was odd. Hence, $x$ must be odd which gives us that $x\in \{n\in {\mathbb N}\,|\, n\mbox{ is odd}.\}$. Hence, we've shown that each set has exactly the same elements which shows that they must be equal sets.


Example. Show that \[\{x\in{\mathbb N}\,|\,x\mbox{ is even.}\} \neq \{2, 4, 8, 16, 32, \dots\}.\]

First, note that it's relatively easy to show these sets aren't equal by showing that there is an element in one which is not in the other; for example, $6$ is in the left-hand set, but not the right-hand set. Despite their inequaity, we might want to find any relations between the sets: note that, for example, the right-hand set is a subset of the left-hand set (why?) and so we have \[\{2, 4, 8, 16, 32, \dots\}\subseteq \{x\in{\mathbb N}\,|\,x\mbox{ is even.}\}.\]


Example. Show \[\{n\in {\mathbb Z}\,|\, n\mbox{ is positive}.\} \neq \{n\in {\mathbb Z}\,|\, n^{2}\mbox{ is positive.}\}.\]

As before, we need to find an element in one set which is not in the other set. Be careful when reading these sets, also: the second set is talking about the elements $n$ such that $n^2$ is positive; hence, the second set is actually all of ${\mathbb Z}$ except for the element $0$ (why is this?). The left-hand set only has the elements $\{1,2,3,4,\dots\}$. Hence, for example, $-1$ is in the left-hand set but not the right hand set. Indeed, it is actually the case that \[\{n\in {\mathbb Z}\,|\, n\mbox{ is positive}.\}\subseteq \{n\in {\mathbb Z}\,|\, n^{2}\mbox{ is positive.}\},\] which you should be able to show.

At this point, we should be able to be clever enough to prove certain properties about sets and their operations. We'll prove one, and then we'll just note some of the rest.

Proposition. Given sets $A,B,C$ we have that $A\cap (B\cup C) = (A\cap B)\cup (A\cap C)$.

This looks a bit intimidating at first glance, but let's go through it like we did the others: we need to show each side is a subset of the other.
 
Proof. Take $x\in A\cap (B\cup C)$; then $x\in A$ and $x\in B\cup C$; this means that it is certain that $x\in A$, and also that $x$ is in either $B$ or $C$ (or perhaps both). Let's suppose that $x\in B$; then $x\in A\cap B$. Hence, it is in $(A\cap B)\cup (A\cap C)$ which is the right-hand side. Good! Note that if we supposed that $x\in C$ instead of $B$ in the sentence before, we get the same conclusion.
 
Now, take $x\in (A\cap B)\cup (A\cap C)$. This implies that it is either the case that $x\in A\cap B$ or $x\in A\cap C$ (or perhaps both). Let's assume $x\in A\cap B$ (the other case, that $x\in A\cap C$, is similar). Then $x\in A$ and $x\in B$ by definition of $\cap$. Since $x\in B$, it is also in $x\in B\cup C$ (why is this?), so we have $x\in A\cap (B\cup C)$. This is the left-hand side of the equality. Hence, these sets are equal to each other.

There's a few other equalities that we might need, and a few that we'll prove on the spot when we need them. These equalities are from the De Morgan's Laws, perhaps because De Morgan formalized them. Note that some of these can be derived from the others, but I'm including them just for easy reference.

For $A,B,C$ sets, we have that... \[(A\cup B)^{C} = A^{C} \cap B^{C}\] \[(A\cap B)^{C} = A^{C} \cup B^{C}\] \[A\cap (B\cup C) = (A\cap B)\cup (A\cap C)\] \[A\cup (B\cap C) = (A\cup B)\cap (A\cup C)\] \[\left(\bigcup_{i} A_{i}\right)^{C} = \bigcap_{i} A_{i}^{C}\] \[\left(\bigcap_{i} A_{i}\right)^{C} = \bigcup_{i} A_{i}^{C}\]

In general, there is a neat duality between $\cup$ and $\cap$ via the complement, but we won't explore this here.
 
There's a number of other topics in set theory but most of them are built off of the foundation we've just created. We will introduce the rest when they become useful for us.

Next time...

In the next post, we'll discuss what a topology is, what it looks like, and what some common properties of spaces are.