The purpose of this post is to serve as a reference I can link to. I define topological spaces, sure, but the goal is really to communicate what they feel like, how I think about them, what I think is neat about them.

What does a topologist do?

The common “joke” about a topologist is that they cannot tell the difference between a doughnut and a coffee cup. One year my brother got me a white mug with text as a gift that reads “this is a doughnut. trust me, i’m a topologist”.

The joke isn’t particularly funny when you first here it because you don’t understand what is true about it. Topology is the mathematical study of continuity, of shape and of space. It is interested in what inheres in a shape if you are allowed to bend or stretch or squash a shape without breaking or tearing it. This sets it against geometry, which is interested in more rigid properties of shapes and spaces, but many geometers are also interested in topology, and vice versa. The reason that a topologist would consider a doughnut and a coffee cup the same is because if you imagined that both were made out of silly putty or some other malleable material, you could transform one into the other by moulding it with your hands without ever having to tear or cut the material.

Topology, like group theory, is a foundational part of mathematics. However, in the U.S., particularly in smaller colleges, not every undergraduate who majors in mathematics studies topology. They may meet notions that are central to the study of topology—continuity of functions, and open and closed sets, for example—in a class like Real Analysis instead.

What is an example of a piece of topology research I can think about?

I realized, in an exercise that I did at a Communicating Math workshop in 2022, that I can explain the concept of an end of a topological space to almost anyone. I loved this discovery, it really transformed my ability and willingness to explain what I think about to people. Forgive me for repeating myself if you’ve heard me give this spiel before (I tried it on almost all of my friends, I did it on Dani Derks’ Sound and Process podcast, even when interviewing with the Dean of the school of Arts and Sciences at Rutgers–Newark for my job as an assistant professor.)

Consider a line—one that goes on forever in both directions. Walk up to this line, and imagine taking a chunk of the line from the middle. Throw that chunk away; it’s gone. How many pieces are left over?

The answer I’m looking for is “two”: the “left” half and the “right” half.

Once we agree that there are two pieces left over, I ask if that fact depended on the size of the chunk that I threw away, and people usually say (and I agree) that no, it does not. For this reason, I explain, topologists say that the line has two ends.

Very occasionally somebody realizes that my use of the word “chunk” is a little underspecified: if you think of two or more discrete chunks when you’re throwing things away, there may be more than two pieces left over. Also, “chunk” has a feeling of smallness that I’m leaning on implicitly. If I threw away the entire left half of the line, for instance, there might be only one piece left over.

To lawyer these difficulties away, one could be more precise: instead of “chunk” I could say “compact” or “closed and bounded” subset (more about these below), and instead of leaning on a notion of “size” or “connectedness”, I could invoke the notion of an “inverse limit”, which will one day have its time to shine here on the blog.

Anyway, I say none of this to my friend if it doesn’t come up. Instead, I point out that if we change the situation, the behavior might change. Think of a plane that goes on forever in all directions instead. Now do the same thing; walk up to the plane and take a chunk out and throw the chunk away. Now people sometimes have different ideas initially about how many pieces are left; the answer I’m looking for here is “one”, and I usually justify it (complete with a little walking hand gesture) by observing that you can walk all the way around the outside of the hole that is left and come back to where you started. So the line has two ends, but the plane has one.

A beautiful little theorem due to Freudenthal and Hopf says that the notion of “ends” makes sense for (abstract) groups as well. There the number of ends of a group is the number of ends of a topological space which it acts on “nicely”. What’s more, although you can come up with shapes that have essentially arbitrary numbers of ends, for example a “Y” shape that goes on forever would have three ends, for a group the possible numbers of ends are “zero”, “one”, “two”, or “infinity”. In other words, if you have at least three ends, you actually have infinitely many.

What is topology formally?

A topological space is a set $X$, together with $\tau$ a collection of subsets of $X$ called a topology on $X$. The subsets making up $\tau$ are called “open” sets, and they must satisfy the following constraints:

• $X$ itself is open, as is the empty subset $\varnothing$ of $X$.
• If $U$ and $V$ are open subsets of $X$, their union (that is, the collection of elements belonging to either $U$ or to $V$) is also open. Actually, the union of any number of open sets is still open.
• If $U$ and $V$ are open subsets of $X$, their intersection (that is, the collection of elements belonging to both $U$ and to $V$) is also open. More generally, the intersection of any finite number of open sets is still open.

I usually think of open sets as being kind of fuzzy; occasionally people will draw them with line boundaries that are somehow soft, perhaps dotted or dashed. When discussing intervals of the real line, for example, an interval will be “open” if it does not contain its endpoints.

The opposite of “open” is “closed”, so in topology a set $S$ in a topological space $X$ is closed if its complement (that is, the collection of elements belonging to $X$ but not to $S$) is open. Closed sets for me have sharp corners.

Many topologies are defined by rules. For example, the open sets of the real line are the sets $U$ for which for any element $x$ in $U$, you can find a little open interval containing $x$ and contained in $U$. So the closed intervals in the real line are not open, because this rule isn’t satisfied for their endpoints.

Every set $S$ has two topologies on it “for free”: the discrete topology in which every subset is open, and the indiscrete topology, in which only the whole set $S$ and the empty subset $\varnothing$ are open. Occasionally different topologies on the same set are relevant, but often it is convenient to pretend that different topological spaces do not have the same elements.

What are some core topological concepts?

Limits

This is a notion that you might remember from Calculus: a limit of a sequence of elements $x_1,x_2,\ldots$ of a set $X$, if it exists, is heuristically a point $x$ that the other points “approach”. It may or may not be the case that some $x_n$ is equal to $x$; what is important is that for any open set $U$ which contains $x$, eventually we can pick a number $N$ so that all of the remaining $x_n$ are also contained in $U$.

In a topological space like the real line that is additionally a metric space, so that we can measure distances, the distance between $x_n$ and $x$ should get smaller and smaller as $n$ gets larger and larger. Formalizing this is the “epsilon–delta” formulation of continuity that many Calculus students struglle with.

Continuity

A function $f \colon X \to Y$ is continuous if preimages of open sets are open. That is, if $U$ is an open subset of $Y$, the collection of all elements of $x$ with the property that $f(x)$ lands in $U$ must be open.

This is pretty different at first blush from the notions of continuity that a student might meet in, say, a calculus class. There what is stressed is that continuous functions play nicely with limits. This is true with our more abstract definition of continuity, and our formulation reveals what is core to topology that makes it different from analysis or Calculus.

Connectedness

Think about the closed intervals $[0,1]$ and $[2,3]$ on the real line. Together they form a subspace $X$ of the real line: the open sets in this subspace are the sets $U \cap X$, where $U$ is open in the real line. For example, $[0,1]$ is not open in the real line, but it is equal to $X \cap (\nicefrac{-1}{2}, \nicefrac{3}{2})$, so it is open in $X$. Since $[0,1]$ is closed in the real line, it is closed in $X$ too. Mathematicians often refer to such subsets with the clunky portmanteau “clopen”.

A topological space $X$ is connected when its only clopen subsets are $X$ and $\varnothing$. In other words, a clopen space has no nonempty proper clopen subsets. The space $X$ from the previous paragraph has two disjoint proper clopen subsets, $[0,1]$ and $[2,3]$. Since these subsets are themselves connected, they form the set of connected components of $X$.

Compactness

Every topological space $X$ has an open cover, a collection of open subsets whose union is all of $X$. A space is compact when every open cover has a finite subcover; that is, if infinitely many open sets cover $X$, actually you only needed finitely many of them to get it done.

The word “compactness” evokes a sense of smallness; think “compact car”, for instance. Many mathematical objects conform to this expectation: circles, closed intervals like $[0,1]$, finite sets, all are compact. A mathematical result named for Heine and Borel says that for subsets of $\mathbb{R}^n$ with the usual topology, a subset $K$ is compact if and only if it is closed, and bounded, in the sense that there is a number $M$ for which two points in $K$ cannot be further than $M$ apart. Some metric spaces (proper metric spaces) satisfy the conclusions of this theorem, but some do not: compact sets are always closed and bounded, but some spaces have sets which are closed and bounded yet not compact.

Compactness is really central to topology, so it’s a little puzzling that its formulation should be so tough to parse on a first go.

Does it have practical applications?

Like group theory, I would argue that topology at this level is foundational to our understanding of other subjects. For example, the “manifold hypothesis” in data analysis posits that if you plot data along many many numerical axes (that is, you think of “embedding” that data in $\mathbb{R}^n$ for some big number $n$), that the data actually belongs to a “nice” kind of topological space called a “manifold” whose “intrinsic dimension” is somehow much smaller than $n$.

Understanding the “shape” of an object is fundamental in many aspects of human endeavor beyond just reasoning about the “shape of data”, which was all the rage seven or eight years ago.

Where can I read more?

I learned topology from a book by that same name by James Munkres. I loved my topology class as a college student; something about it just spoke my language, and I earned what may have been my only “A+” grade in college in that class.

Like the group theory post, this post forms a link at the bottom of my chain.