Explain the general idea of topology for a sophomore student

Question

I am a sophomore and still taking calculus 2 and 3. However, I asked several questions in class but the professor always answers me: you need to take a higher level of mathematics, topology in specific, to be able to answer these questions.

For example, when we were taking how to derive and integrate functions with multiple variables, I noticed that both differentiation and integration are generally expressed in terms of limits. In fact, this was a shock for me because in high school I used to treat derivatives, integrals, and limits such as every one of them have its own rules and I thought that they are somehow independent. It was the first time I see their proofs and understand what they really mean. So I've just wondered and asked if there is anything more general that can define limits. The professor answered: topology. Topology again! It was the third time this semester I received the same mysterious answer.

Therefore, I decided to search and read some books related to topology. However, most explanations and books require a higher level of mathematics than I have.

Can anyone explain the general idea of topology and recommend some simple books?

Answer 1

The ideas of topology were first encountered by people studying limits and geometry, but they run much more broadly and deeply. I like to think of topology as a first step beyond logic. I learned this perspective from an excellent post on the blog XOR's Hammer, but you might prefer the gentler introduction below.

Propositions and subsets

Logic is the study of propositions that can be true or false:

• "This team will win its first four games."
• "This team will lose at least one of its first eight games."
• "This team will lose a game someday."
• "This team will win every game it ever plays."

It's often useful to look at propositions as subsets of a set.

• $X$: a set of teams.
• $W_n \subset X$: the subset of teams that win their first $n$ games.
• $V_n \subset X$: the subset of teams that lose at least one of their first $n$ games.
• $U \subset X$: the set of teams that lose a game someday.
• $A \subset X$: the subset of teams that win every game they ever play.

This point of view makes logical operations look like set operations.

• $W_4 \cap U$ (an intersection of two subsets): "This team will win its first four games and lose a game someday."
• $W_4 \cup V_4$ (a union of two subsets): "This team will win its first four games or lose at least one of its first four games."
• $\neg U$ (the complement of a subset): "This team will never lose a game."
• $W_1 \cap W_2 \cap W_3 \cap \ldots$ (an intersection of an infinite family of subsets): "This team will win its first game, and its first two games, and its first three games, and..."

It also turns relationships between propositions into relationships between subsets.

• $X = W_4 \cup V_4$: Every team will win its first four games or lose at least one of its first four games.
• $A = \neg U$: Always winning is the same as never losing.
• $A = W_1 \cap W_2 \cap W_3 \cap \ldots$: Winning every game just means winning the first game, and the first two games, and the first three games, and so on...

Talking about verifiable propositions

It seems reasonable to bet that a team will win its first four games, or that it will lose at least one of its first eight games. If your bet succeeds, you'll find out pretty soon. It could even be reasonable to bet that a team will lose a game someday. If your bet succeeds, you'll find out someday.

On the other hand, it doesn't seem reasonable to bet that a team will win every game it ever plays. Even if your bet succeeds, there will never be a point when you know it's succeeded. You'll never collect your winnings!

By thinking about what our propositions are supposed to mean, we've assigned some of them the special property of being verifiable. From a logical point of view, our choice of which propositions to call verifiable was sort of arbitrary: it was based on our interpretation of the propositions, not on the inherent logical relationships between them. However, our intuitive understanding of what verifiable is supposed to mean might give us some expectations about how verifiable propositions should relate to each other. For example:

• If we successfully bet that "this team will win its first four games and lose a game someday," we can collect our winnings as soon as we've seen the first four games and we've seen the team lose. Abstractly, if we successfully bet on one verifiable proposition and another, we'll know the whole bet succeeds as soon as we know both parts succeed.
• If we successfully bet that "this team will lose its first game, or at least one of its first two games, or at least one of its first three games, and so on...," we can collect our winnings as soon as we see the team lose. Abstractly, if we successfully bet on the or of a family of verifiable propositions, we'll know the whole bet succeeds as soon as we know one of its parts succeeds. This works even if the family is infinite!
• (On the other hand, if we successfully bet that "this team will win its first game, and its first two games, and its first three games, and so on...," we'll never collect our winnings. This shows that the and of an infinite family of verifiable propositions isn't guaranteed to be verifiable.)
• There are some propositions, like "this team will lose a game someday or win every game it ever plays," which are sure to be true, just on logical grounds. If we bet (successfully, of course) on a sure thing, we know immediately that the bet succeeds.
• There are some propositions, like "this team will win its first four games and lose at least one of its first four games," which are sure to be false, just on logical grounds. If we bet successfully on something that's sure to be false—well, that won't happen. So we can technically say, without lying, that if it happened, we'd find out someday that the bet succeeds.

Leaving the examples and reasoning behind, we end up with a few general expectations for how the propositions we choose to call verifiable should relate to each other:

• The and of two verifiable propositions is verifiable.
• The or of any family of verifiable propositions is verifiable.
• A proposition that's sure to be true, or sure to be false, is verifiable.

Verifiable propositions as subsets

Let's translate these expectations from the language of logic to the language of subsets. Logical operations, like and and or, become set operations, like intersection and union. The word verifiable feels tied to logic, since it evokes the idea of truth. Let's replace it with the word open, which has less baggage.

• The intersection of two open subsets is open.
• The union of any family of open subsets is open.
• The empty subset $\varnothing \subset X$ and the full subset $X \subset X$ are both open.

When we choose which subsets of a set $X$ to call open, making sure to fulfill the expectations above, we've given $X$ what mathematicians call a topology.

Many sets come with a traditionally standard topology. You already know one example, because I've been using it from the beginning.

In the standard topology on the set of infinite binary sequences, the following subsets are considered to be open.

• Any subset you get by specifying the first $n$ digits of the sequence (for some integer $n \ge 0$) and letting the rest of the digits vary.
• The union of any family of such subsets.

Here's another example that might feel familiar to you.

In the standard topology on the set of real numbers, the following subsets are considered to be open.

• Any open interval $(a, b)$.
• The union of any family of such subsets.

(In both examples, the empty subset is considered open because it's the union of the empty family.)

Answer 2

What follows is very informal. To prescribe a topology to a set is to provide a rule to decide what are the "neighbourhoods" of the points of the set.

While it may be tempting to view the topology of a space as a means of measuring the proximity between different points in the space, it's actually more general to consider it as a formal method for determining whether a point is close to a particular subset of the space rather than to other points in the space. In this sense, a point is considered "close" to a subset if every open neighbourhood (according to the specified topology) that contains the point intersects the aforementioned subset.

At the end, this local information can be "glued together" to obtain global information about the shape of the space. For example, the interval $[0,2\pi)$ can be endowed by the usual topology, and this way it adquires its "usual shape", like a segment, a piece of line... But you can prescribe another topology, in which "sets near $2\pi$" are also considered to be "near to $0$", and then the shape of this new topological space is that of $S^1$.

There are very nice topological spaces, like $\mathbb R^n$ with the usual topology induced by a metric, where you have enough "precision" even to create calculus: limit, derivatives, and so on.

Added after the other answer:

Of course metric spaces are a good source of examples of topologies, in particular the real line with its usual topology, and they help to construct intuition, but I think is a bad idea to spend "too much time" in Real Analysis with the goal of "feeling what Topology is". The ideal point, I think, is to study general topology while translating the definitions and results to a metric space. The particular case of the real line is important, but focusing only in it always was misleading to me, because it is too well known and "too thin" to see the ideas.

Answer 3

I would suggest (at least temporarily) avoiding Topology, and instead looking for a somewhat proof-oriented textbook on Real Analysis. One example is "Calculus" (volumes I and II), 2nd Ed. (Tom Apostol).

However, this may not be the right book for you. You have to find a proof-oriented Real Analysis book, with a lot of exercises, that dovetails with your interests, Math education, and Math ability. You want the book to (only moderately) stretch your intution.

Depending on the nature of your questions, you may be able to resolve all of the questions that you are curious about via the right Real Analysis book. If so, this would allow you to avoid a deeper study of Topology. This avoidance is important if Topology does not happen to be something that you are independently interested in.