It's been years since my last (a.k.a. first) post, where I described a presentation of some basic concepts in point set topology in terms of a concept I called "touching", though I believe that it also shows up in the literature under the name nearness.
Since writing that post, I've occasionally revisited this conception of topology, and revisited my old grad school textbook (Munkres' Topology, Second Edition), re-interpreting the text through the lens of touching. I've found the experience quite enlightening: I feel like I keep acquiring more comfort and intuition around topology, moreso than I would just reading things as-is. Along the way I've written what looks like a bunch of notes, but they're in my chicken-scratch handwriting and relegated to a folder of notes on my bookshelf. Rather than let those explorations yellow and decay, I'd like to share some of them here in case they're helpful to others. Plus they give me a chance to clarify my thinking and maybe get some corrections or insights from others.
A word of warning: reading these notes is probably not the way to learn topology. I'll only provide so much motivation, and things will not necessarily arise in the same order as Munkres' textbook, nor in an order that explains why a given concept or theorem arises. I'll try to leave some breadcrumbs or pointers into Munkres at least. Suggestions/questions/comments welcome!
For a quick review, in the last post [link?] I introduced the idea of a topological space as a set \(\U\), the universe, paired with an binary relation \(x \touches A\) pronounced "\(x\) touches \(A\)" that relates an element \(x \in \U\) to a subset \(A \subseteq \U\) and satisfies some axioms:
- \(x \ntouches {\emptyset} \)
- If \(x \in A\) then \(x \touches A\)
- If \(x \touches A\) and \(A = B \cup C\) then \(x \touches B\) or \(x \touches C\)
- If \(x \touches A\) and \(a \touches B\) for all \(a \in A\), then \(x \touches B\)
I gave some intuition for the axioms, which lean heavily on how we think about space.
Then I introduced the idea of continuous functions, which is a common topic in calculus, but we generalized it to this abstract notion of touching. However the notion of continuous function from calculus is just a special case of the topological one for a particular topological space defined over the set of real numbers, or pairs of real numbers in 2-dimensions, or triples of real numbers in 3-dimensions, and so on. We used the 1-dimensional version in the last post, but it generalizes. We can call this the standard topology over Euclidean space (which is named after Euclid, the fellow who documented the axioms of (Euclidean) geometry that have influence mathematics for many many centuries).
I also introduced some common topological concepts, which give some general, though abstract, definitions to terminology that we might know from geometry or from talking about the world around us. Terms like interior, boundary, and friends are applied more broadly to worlds where touching is given a loose interpretation in terms of these axioms.
To continue, let's introduce some more topological terminology/concepts, and definitions for those. This stuff comes up a lot as the topic develops, and it's fun (if you're like me) to think about the intuition behind the definitions. In fact, I recall having some fun figuring out how to translate more standard definitions into touching-based definitions that fit my intuitions well. I'm a strong believer in the importance of intuition in mathematics. Two characterizations of a concept may be equivalent as far as mathematics goes (in the sense that any statement that you can prove using one can also be proved using another), but the effect of intuition on the human mind and how it helps you learn more mathematics or figure out how to apply a concept (or modify it to suit a different problem, or develop another concept by analogy) is priceless. For me, as I've already said, the definitions in terms of open sets, which are standard, don't connect well with my intuitions, and talking to others I know I'm not alone. Maybe these alternate definitions will help.
Connectedness
Here's another core concept: let's define what it means for a topological space \(\U\) to be connected. As Munkres explains at the beginning of Chapter 3, connectedness is a significant topological property of closed intervals of real numbers (when given the topology that we implicitly and intuitively associate with real numbers: see below). A famous theorem in calculus, the intermediate value theorem, can be proven in an abstract way within topology, and helps clarify which properties of real numbers and continuous functions really make the theorem go through. Remember that topology abstracts the concept of continuous functions away from numbers, so that it can be applied to other topological spaces, but some theorems that were originally considered in calculus can still be stated and proven. So topology makes it possible to capture properties of mathematical objects and functions that are fundamentally about their topology (about the relevant notions of touching and the continuous functions that arise from them). Some facts of calculus go beyond topology, for instance if they are fundamentally about distance (between points) and not just about touching. This is the sense in which topology and calculus are not the same topic of study, but many theorems in calculus generalize to theorems about any topological space that shares enough in common with the real numbers given their standard topology.
Anyway, let me start to explain connectedness with an analogy: my childhood home. Upon arriving home in a car, I would have to exit the garage, go around the side of the house and then enter through the back door, which led to the laundry room. Once there, I could walk from any room of the house into any other room of the house except for the garage. If I wanted to get to the garage, I'd have to go outside and then in through the garage door. So in my mind, the garage was not "connected" to the inside of the rest of the house, even though the garage was attached by walls to the house. You could say that here I'm focused on the "topology" of the "walkable space" of my home, rather than the "topology" of the blueprints, walls and all.
Now let's introduce the formal definition of connectedness and then relate it to my childhood home. A space \(\U\) is connected if and only if:
[ed, May 11, 2023: previously I got this definition wrong: there was a \(\wedge\) where the \(\vee\) now is. Ack!]
That's a mouthful! What does it say? Let's take it piece by piece and interpret it as literally as we can. It says that the entire space \(\U\) is connected if every subset \(A\) that is neither empty (\(\exists x \in \U.\,x \in A\)) nor the entire universe (\(\exists x \in \U.\,x \in \inv{A}\)) has an element that touches the outside (\(\exists a \in A.\, a \touches (\U \setdiff A)\)) or is touched by some element from the outside (\(\exists x \in (\U \setdiff A).\, x \touches A\)).
Phew! Now for a more colloquial interpretation. No matter which way we split the universe into two non-empty sets, \(A\) and \(\inv{A}\), we can find an element of one of those sets that touches the other. So we're taking this notion of touching as a means to get from one side of the universe to the other!
Going back to my house, we can argue that my garage is not connected to the rest of the house, because if we take the walkable part of the garage as one set and the walkable part of the rest of the house as another set (it does not matter which one is \(A\) and which one is its inverse \(\inv{A}\)) then neither touches the other in that there is no entry "point" from one that touches anywhere in the other.
Things can get a bit more subtle though. If I had lived in a strange castle with a trap door into the dungeon but no stairs for getting out, then our definition of connectedness would say that the dungeon is connected to the rest of the castle, even though I could not freely move back and forth.
Mind you, this example is just an analogy to get you thinking about the idea of connectedness, but we can also consider some actual mathematical examples.
First, an example of a connected space is a closed range of numbers:
Let \( \U = [5,10] = \lbrace x \in \R \mid 5 \leq x \leq 10\rbrace \), and \(x \touches A\) if and only if \( \mathit{glb} \{ \lvert a - x \rvert \mid a \in A \} = 0. \)
We saw a similar definition of touching in our last episode, which in words says that \(x\) touches \(A\) if the greatest lower bound on distance between \(x\) and any element of \(A\) is zero. This notion of touching is super pervasive when dealing with numbers, be they rational, real, and can be extended to tuples of numbers too with a little work. For now I'll call it the standard definition, though I'll only use it with subsets of real numbers for now.
Anyway, this space, which is a contiguous range of real numbers, is connected. No matter how we split it in two, there will always be some element in each set that touches the other. Proving this involves considering all possible splits, and showing that there must always be at least one element that touches across the divide. And one can come up with a lot of crazy ways to allocate elements of [5,10] into 2 sets.
For instance, you could split the set into the (infinite) set of rational numbers (i.e. numbers that can be written as a ratio of natural numbers) between 5 and 10 and the (infinite) set of irrational numbers (numbers that cannot be written as ratios) between 5 and 10. For this particular split, I would pick some rational number in the middle, say 7, and prove that it touches the set of irrational numbers between 5 and 7, and thus touches the set of irrational numbers between 5 and 10 (since adding more stuff to a touched set preserves touching). But you want to prove this for all possible splits. Check out Munkres for more on how to really do it: this discussion is just to get you thinking about what you would need to prove in general using one particular (nutty) split.
For our second example, consider the set \(\U = [1,4] \cup [5,10]\), which combines two closed intervals: all the numbers from 1 to 4, and all the numbers from 5 to 10, and use the standard definition of touching for numbers. It is much simpler to show that this set is not connected than it was to show that the previous example was connected: we just need to exhibit one split and show that for each of the two sets, every element of one set does not touch the other. We split this one set into the two closed intervals, and pick an element \(a \in [1,4]\). Now we can show that \(a \ntouches [5,10]\): assume that \(a \touches [5,10]\) and prove that this implies a contradiction (to be clear, this strategy is proof of a negation, not proof by contradiction):
2. Then by definition of touching, 0 is the greatest lower bound of distances between \(a\) and any element of \([5,10]\);
3. But since 4 is the largest number that \(a\) could be, then \(5-4 = 1\) is a lowerbound of any \(a \in [1,4]\) distance.
We can play the same game in the other direction to complete the proof.
There's plenty more to the concept of connectedness. For instance, why is it that an entire space is connected, rather than some subset of a topological space? Well, there is a way to take the former concept and use it to define the latter, but we don't have enough tools yet (in particular, how to define a subspace of a topological space), but we can get there once we do :).
Parting Quiz Question: take \(\U = [5,10]\) but take touching to be \(a \touches A\) if and only if \(a \in A\). In other words, assume the discrete topology (introduced last post). Is this topological space connected? Why or why not?
First Posted October 25, 2022