If you’ve got a connected topology, there are some neat things you can show about it. One of the interesting ones involves *fixed points*. Today I’m going to show you a few of the relatively simple fixed point properties of basic connected topologies.
To give you a taste of what’s coming: imagine that you have two sheets of graph paper, with the edges numbered with a coordinate system. So you can easily identify any point on the sheet of paper. Take one sheet, and lay it flat on the table. Take the *second* sheet, and crumple it up into a little ball. No matter how you crumple the paper into a ball, no matter where you put it down on the uncrumpled sheet, there will be at least one point on the crumpled ball of paper which is directly above the point with the same coordinate on the flat sheet.
As a quick aside: today is Yom Kippur, which means that this post is scheduled, and I’m not anywhere near my computer. So I won’t be able to make any corrections or answer any comments until late this evening.
Next stop on our tour of topology is the idea of *connectedness*. It’s an important concept that defines a lot of useful and interesting properties of topological spaces.
The basic idea of connectedness is very simple and intuitive. If you think of a topology on a metric space like ℜ3, what connectedness means is, quite literally, connectedness in the physical sense: a space is connected if doesn’t consist of two or more pieces that never touch.
Being more formal, there are several equivalent definitions:
* The most common one is the definition in terms of open and closed sets. It’s precise, concise, and formal; but it doesn’t have a huge amount of intuitive value. A topological space **T** is connected if/f the only two sets in **T** that are both open *and* closed are **T** and ∅.
* The most intuitive one is the simplest set based definition: a topological space **T** connected if/f **T** is *not* the union of two disjoint non-empty closed sets.
* One that’s clever, in that it’s got both formality and intuition: **T** is connected if the only sets in **T** with empty boundaries are **T** and ∅.
Closely related to the ida of connectedness is separation. A topological space is *separated* if/f it’s not connected. (Profound, huh?)
Separateness becomes important when we talk about *subspaces*, because it’s much easier to define when subspaces are *separated*; and they’re connected if they’re not separated.
If A and B are subspaces of a topological space **T**, then they’re *separated in **T*** if and only if they are disjoint from each others closure. An important thing to understand here is that we are *not* saying that their *closures* are disjoint. We’re saying that A and B* are disjoint, and B and A* are disjoint, not that A* and B* are disjoint.
The distinction is much clearer with an example. Let’s look at the topological space ℜ2. We can have *A* and *B* be *open* circles. Let’s say that *A* is the open circle centered on (-1,0) with radius one; so it’s every point whose distance from (-1,0) is *less than* 1. And let’s say that *B* is the open circle centered on (1,0), with radius 1. So the two sets are what you see in the image below. *A* is the shaded part of the green circle. The outline is the *boundary* of *A*, which is part of *A**, but not part of *A* itself. *B* is the shaded part of the red circle; the outline is the boundary. Neither *A* nor *B* include the point (0,0). But both the *closure* of *A* and the *closure* of *B* contain (0,0). So *A* is disjoint from *B**; and *B* is disjoint from *A**. But *A** is *not* disjoint from *B**: they overlap at (0,0). *A* and *B* are separated, even though their closures overlap.
There’s also an even stronger notion of connectness for a topological space (or for subspaces): *path*-connectedness. A space **T** is *path connected* if/f for any two points x,y ∈ **T**, there is a *continuous path* between x and y. Of course, we need to be a bit more formal than that; what’s a continuous path?
There is a continuous path from x to y in **T** if there is a continuous function *f* from the closed interval [0,1] to **T**, where *f(0)=x*, and *f(1)=y*.
This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it’s such a useful tool for talking about other things. Well, today, I’m going to show you a great example of that.
Last friday, I went through a fairly traditional approach to describing the topological product. The traditional approach not *very* difficult, but it’s not particularly easy to follow either. The construction isn’t really that difficult, but it’s not easy to work out just what it all really means.
There is another approach to presenting it using category theory, and to me at least, it makes it a *whole* lot easier to grasp. To make the diagrams easier to draw, I’ll adopt one shorthand: instead of writing (T,τ) for topological spaces, I’ll use a single symbol, like **X**, with the understanding that **X** represents the *pair* of the set and the topology that form the topological space.
Suppose we have a set topological spaces, **E**1, **E**2, …, **E**n. The product **P** = Πi=1..n**E**i is the *only* topological space with projection functions pi : **P** → **E**i, such that
for any other topological spaces **S**, if **S** has continuous functions fi : **S** → **E**i to each of the elements of the product, then there is *exactly one* continuous function g : **S** → **P** such that the following diagram commutes:
That’s really just a repetition of the definition of categorical product, just made specific to the category **Top**. Everything I said in fridays post about what forms the open sets of the topological product space is directly implied by this categorical definition. The property of the open sets of the product topology being the coarsest structure of sets that maintains the structural properties of the product element topologies – that’s implied by the categorical description.
To me, this is the real beauty of category theory, and the whole reason why I spent all that time explaining it. Being able to describe structures in the language of category theory makes things much easier to understand.
One of the really neat things you can do in topology is play games with dimensions. Topology can give you ways of measuring dimensions, and projecting structures with many dimensions into lower-dimensional spaces. One of the keys to doing this is understanding how to combine different topologies to create new structures. This is done using the *topological product*.
Just like you can define a sub-set of a set, or a sub-object of an object in a category, you can define a sub-*space* of a topological space. It’s a pretty easy thing to understand; interestingly, a sub-space of a topological space works in pretty much exactly the same way as a sub-sets and sub-object. In fact, the topological definition of a sub-space is *identical* to the categorical definition of a sub-object when we’re looking at the category of topologies, **Top**.
Today, I’m going to explain what a subspace is, and show you how the categorical sub-object corresponds to the topological subspace. Read on beneath the fold.
When we talk about topology, in general, the way we talk about it is in terms of *shapes*: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a *torus*, or a *coffee mug*, or a *sphere*.
But the topology we’ve talked about so far doesn’t talk about shapes or surfaces. It talks about open sets and closed sets, about neighborhoods, even about filters; but we haven’t touched on how this relates to our *intuitive* notion of shape.
Today, we’ll make a start on the idea of surface and shape by defining what *interior* and *boundary* mean in a topological space.
The past couple of posts on continuity and homeomorphism actually glossed over one really important point. I’m actually surprised no one called me on it; either you guys have learned to trust me, or else no one is reading this.
What I skimmed past is what a *neighborhood* is. The intuition for a
neighborhood is based on metric spaces: in a metric space, the neighborhood of a
point p is the points that are *close to* p, where close to is defined in terms of the distance metric. But not all topological spaces are metric spaces. So what’s a neighborhood in a non-metric topological space?
With continuity under our belts (albeit with some bumps along the way), we can look at something that many people consider *the* central concept of topology: homeomorphisms.
A homeomorphism is what defines the topological concept of *equivalence*. Remember the clay mug/torus metaphor from from my introduction: in topology, two topological spaces are equivalent if they can be bent, stretched, smushed, twisted, or glued to form the same shape *without* tearing.
The rest is beneath the fold.
*(Note: in the original version of this, I made an absolutely **huge** error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in terms of the *inverse* of the function; I originally wrote it in the other direction. Thanks to commenter Dave Glasser for pointing out my error. I’ll try to be more careful in the future!)*
Since I’m back, it’s time to get back to topology!
I’m going to spend a bit more time talking about what continuity means; it’s a really important concept in topology, and I don’t think I did a particularly good job at explaining it in my first attempt.
Continuity is a concept of a certain kind of *smoothness*. In non-topological mathematics, we define continuity with a very straightforward algebraic idea of smoothness. A standard intuitive definition of a *continuous function* in algebra is “a function whose graph can be drawn without lifting your pencil”. The topological idea of continuity is very much the same kind of thing – but since a topological space is just a set with some additional structure, the definition of continuity has to be generalized to the structure of topologies.
The closest we can get to the algebraic intuition is to talk about *neighborhoods*. We’ll define them more precisely in a moment, but first we’ll just talk intuitively. Neighborhoods only exist in topological metric spaces, since they end up being defined in terms of distance; but they’ll give us the intuition that we can build on.
Let’s look at two topological spaces, **S** and **T**, and a function f : **S** → **T** (that is, a function from *points* in **S** to *points* in **T**). What does it mean for f to be continuous? What does *smoothness* mean in this context?
Suppose we’ve got a point, *s* ∈ **S**. Then f(*s*) ∈ **T**. If f is continuous, then for any point p in **T** *close to f(s)*, f-1(p) will be *close to* *s*. What does close to mean? Pick any distance – any *neighborhood* N(f(s)) in **T** – no matter how small; there will be a corresponding neighborhood of M(*s*) around s in **S** so that for all points p in N(f(s)), f-1 will be in M(*s*). If that’s a bit hard to follow, a diagram might help:
To be a bit more precise: let’s define a neighborhood. A neighborhood N(p) of a point p is a set of points that are *close to* p. We’ll leave the precise definition of *close to* open, but you can think of it as being within a real-number distance in a metric space. (*close to* for the sake of continuity is definable for any topological space, but it can be a strange concept of close to.)
The function f is continuous if and only if for all points f(s) ∈ **T**, for all neighborhoods N(f(s)) of f(s), there is some neighborhood M(s) in **S** so that f(M(s)) ⊆ N(f(s)). Note that this is for *all* neighborhoods of *all* points in **T** mapped to by f – so no matter how small you shrink the neighborhood around f(s), the property holds – and it implies that as the neighborhood in **T** shrinks, so does the corresponding neighborhood in **S**, until you reach the single points f(s) and s.
Why does this imply *smoothness*? It means that you can’t find a set of points in the range of f in **T** that are close together, but that weren’t close together in **S** before being mapped by f. f won’t put things together that weren’t together originally. And it won’t pull things apart that weren’t
close together originally. *(This paragraph was corrected to be more clear based on comments from Daniel Martin.)*
For a neat exercise: go back to the category theory articles, where we defined *initial* and *final* objects in a category. There are corresponding notions of *initial* and *final* topologies in a topological space for a set. The definitions are basically the same as in category theory – the arrows from the initial object are the *continuous functions* from the topological space, etc.
Yesterday, I introduced the idea of a *metric space*, and then used it to define *open* and *closed* sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open sets equivalent to the definition of closed sets. It’s been corrected, but if you’re not familiar with this stuff, you might want to go back and take a look at the corrected version. It’s just replacing a ≤ with a <, but that makes a *big* difference in meaning!)
Today I’m going to explain what a *topological space* is, and what *continuity* means in topology. (For another take on continuity, head on over to [Antopology][antopology] where Ofer has posted his explanation.)
A *topological space* is a set **X** and a collection **T** of subsets of **X**, where the following conditions hold:
1. ∅ ∈ **T** and **X** ∈ **T*.
2. ∀ C ∈ ℘(**T**); : ∪(c ∈ C) ∈ **T**. (That is, the union of any collection of subsets of **T** is an element of **T**. )
3. ∀ s, t ∈ **T** : s ∩ t ∈ T. *(The intersection of any two subsets of **T** is also in **T**.)
The collection **T** is called a *topology* on **T**. The *members* of **T** are the *open sets* of the topology. The *closed sets* are the set complements of the members of **T**. Finally, the *elements* of the topological space **X** are called *points*.
The connection to metric spaces should be pretty obvious. The way we built up open and closed sets over a metric space can be used to produce topologies. The properties we worked out for the open and closed sets are exactly the properties that are *required* of the open and closed sets of the topology. There are many ways to build a topology other than starting with a metric space, but that’s definitely the easiest way.
One of the most important ideas in topology is the notion of *continuity*. In some sense, it’s *the* fundamental abstraction of topology. We can now define it.
A *function* from topological space **X** to topological space **U** is *continuous* if/f for every open sets C ∈ **T** the *inverse image* of f on C is an open set. The inverse image is the set of points x in **X** where f(x) ∈ C.
That’s a bit difficult to grasp. What it’s really capturing is that there are no *gaps* in the function. If there were a gap, then the open spaces would no longer be open. Think of the metric spaces idea of open sets. Imagine an open set with a cube cut out of the middle. It’s definitely not continuous. If you took a function on that open set, and its inverse image was the set with the cube cut out, then the function is not smoothly mapping from the open set to the other topological space. It’s mapping *part of* the open set, leaving a big ugly gap.
Now, here’s were it gets kind of nifty. The set of of topological spaces and continuous functions form a *category*. with the spaces as objects and the functions as morphisms. We call this category **Top**. It’s often easiest to talk about topological spaces using the constructs of category theory.
So, for example, one of the most important ideas in topology is *homeomorphism*. A homeomorphism is a bicontinuous bijection (a bicontinuous function is a continuous function with a continuous inverse; a bijection is a bidirectional total function between sets.) A homeomorphism between topological spaces is a *homomorphism* in **Top**.
From the perspective of topology, any two topological spaces with a homeomorphism between them are *identical*. (Which ends up corresponding exactly to how we defined the idea of *equality* in category theory.)
[antopology]: http://antopology.blogspot.com/2006/08/continuity-introduced.html
