What Are the Key Concepts and Applications of Topology?

[devious_]

What exactly is topology? I know it's used a lot in modern physics, but what other applications does it have?

Now, a little bit on the theoretical side, what's difference between point-set, algebraic, geometric and differential topology? Can anyone provide an example problem on each?

What are the prerequisites to learning topology? And, finally, what level is it considered to be (i.e. over-ambitious high school, undergrad or grad)?

 

[matt grime]

In the US, topology is probably grad, or advanced undergrad.

It is the study of topological spaces, that is spaces that possesses a topology, we do'nt need to explain what those are yet.

the different flavours (algebraic, differential etc) tell you what subclass, if you like, of topologicl space you study and the techniques you will 'use'.

For instance classifying a topological space in a sense we can make precise is very difficult, in fact not possible with the current techniques. You give me two descriptions of a topological space and I almost certainly won't be able to tell you if they are essentially the same. but there are ways of telling them apart and in certain cases that data is sufficient.

One way to do this is to try and find a useful way of assigning a group or ring to the space. that is algebraic topology.

you can also look at continuously differentiable functions on the space, and that is differential topology.

obviously there's more to it than that.

here's an example or two.

Imagine an annulus, that is a circular disc with a hole in the middle, like a cd. pick a distinguished point on the surface. what kinds of paths can you take on the surface that start and end at that point? roughly speaking you can loop once round the hole, or twice round the hole and so on, and you can loop the other way, that's like minus 1 loop round the hole. given a loop, you can then do another loop afterwards, so looping twice round, then going another three times round is like going 5 times round. So, to this punctured disk i can assign the integers in as a group that describes the loops i can make.

If there were two holes i could loop around them in more complicated ways, and there you get ZxZ two copies of the integers.

this is called the fundamental group, it almost, but not quite, measures the number of holes.

that's an example of algebraic topology.

i can't think of an elementary hand waving example for differential geometry, but I'm not a differential geometer.


Let me explain roughly what a topological space is:

It is a set X, with a collection of distinguished subsets of X, often denoted T. So (X,T) is a topological space. the topology is T, and one set may have many different kinds of topologies on it. the elements in T are called *open*, if t is in T, then t^c, the complement of t is called a closed set. The collection of subsets must satisfy certain properties, not just any old collection will do.

point set topology is traditionally just the study of an abstract topological space like this and its topology without any other baggage.

if you'd like the full whack of details say so, but hopefully this has given you an idea of what's iinvolved

 

[devious_]

but hopefully this has given you an idea of what's iinvolved

It has.

Thanks a lot.

 

[HallsofIvy]

Essentially, topology is the abstract study of limits and continuity.

 

[fourier jr]

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects

http://mathworld.wolfram.com/Topology.html

Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness

http://mathworld.wolfram.com/Point-SetTopology.html

The discipline of algebraic topology is popularly known as "rubber-sheet geometry" and can also be viewed as the study of disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of hole structures, and it gets the prefix "algebraic" since many hole structures are represented best by algebraic objects like groups and rings

http://mathworld.wolfram.com/AlgebraicTopology.html

 

[sparkster]

For instance classifying a topological space in a sense we can make precise is very difficult, in fact not possible with the current techniques. You give me two descriptions of a topological space and I almost certainly won't be able to tell you if they are essentially the same. but there are ways of telling them apart and in certain cases that data is sufficient.

Aren't two topolgies the same if they have the same basis? Also, topolgies can be judged to be coarser or finer or noncomparable than each other by looking at their open sets.

 

[Hurkyl]

Here's an example of what matt grime means:

Suppose I told you S is the set of all complex projective solutions to the equation y^2 z = x^3 - x z^2.

What kind of shape does S have? Is it homeomorphic to a sphere? A torus? A Klein Bottle? A hole in a hole in a hole?

(the answer is torus)

 

[sparkster]

Here's an example of what matt grime means:

Suppose I told you S is the set of all complex solutions to the equation y^2 = x^3 - x.

What kind of shape does S have? Is it homeomorphic to a sphere? A torus? A Klein Bottle? A hole in a hole in a hole?

(the answer is torus)

I'm still relatively new to topology, so maybe this question is irrelevent, but if S is as given, what would the open sets be? Or is the set the set of complex numbers and S is what the open sets are?

 

[Hurkyl]

If S is a subset of a topological space T, then S inherits the topology from T. A set is open in S iff it is the intersection of an open set of T with S.


As a simple example, take the interval [0, 1) in R. This is a subset of R, so it inherits the topology of R. A set is open in [0, 1) iff it is the intersection of an open set in R with [0, 1). So, for example, [0, 1/2) is an open set in [0, 1) because it is the intersection of (-1/2, 1/2) with [0, 1).



Oh, note that I changed the problem from affine 2-space to projective 2-space... I almost forgot that you have to include the "point at infinity" to get the isomorphism with the torus.

 

[sparkster]

If S is a subset of a topological space T, then S inherits the topology from T. A set is open in S iff it is the intersection of an open set of T with S.


As a simple example, take the interval [0, 1) in R. This is a subset of R, so it inherits the topology of R. A set is open in [0, 1) iff it is the intersection of an open set in R with [0, 1). So, for example, [0, 1/2) is an open set in [0, 1) because it is the intersection of (-1/2, 1/2) with [0, 1).

But you still have to define the open sets, right? For example, in the regular topology on the real line, [0,1) isn't open. But in lower limit topology it is. All I was asking is what are the open sets in the topology that you're talking about on the complex numbers.

 

[Hurkyl]

Definition: Let (X, O) be a topological space, and let S \subseteq X. Define O|_S := \{ U \cap S \, | \, U \in O \}. Then, the pair (S, O|_S) is a topological space, called the inherited topology from (X, O), or the restriction of (X, O) to S. (Someone correct my phrasing please!)


In other words, a set is open in the inherited topology (also said to be open in S or open relative to S) iff it is the intersection of S with an open set from the parent topology.

Equivalently, a set V is said to be open in S iff every point P of V has a neighborhood U such that U \cap S \subseteq V.


For the solutions to y^2 = x^3 - x, I'm giving it the topology it inherits from \mathbb{P}^2(\mathbb{C}). (Which, incidentally, is the same topology as giving it the inherited topology from \mathbb{C}^2 and then compactifying by adding a single point at infinity)



Hrm, I guess there is a more constructive way to specify the topology: let S be the set of solutions to y^2 = x^3 - x, adjoined with a single point at infinity. That is,

<br /> S := \{ (x, y) \in \mathbb{C}^2 \, | \, y^2 = x^3 - x \} \cup \{ \infty \}<br />

Then, the neighborhoods of S are given by:

<br /> N(P, \epsilon) := \{ Q \in \mathbb{C}^2 \cap S \, | \, ||P - Q|| &lt; \epsilon \}<br />

where P \in \mathbb{C}^2 \cap S and \epsilon &gt; 0 and by

<br /> N(\infty, M) := \{\infty \} \cup \{ Q \in \mathbb{C}^2 \cap S \, | \, ||Q|| &gt; M \}

where M &gt; 0.

 

[Chrono]

I was just reading about topology last night. Let's see if I can recall what I read. Henri Poincare was the guy that came across it. Continuity is what I read about. Squares and circles are the same to a topologists, since they can deform into each other.

 

[fourier jr]

What are the prerequisites to learning topology? And, finally, what level is it considered to be (i.e. over-ambitious high school, undergrad or grad)?

At my school the prereq is either the beginner's analysis course (sets, limits, etc) or advanced calculus. It's probably at the ambitious-undergrad level since its number at my school is 465, but it's also cross-listed as 5-something for grad students.

Can anyone provide an example problem on each?

these probably won't make any sense to anyone, unless they've seen them before
1. Urysohn metrization theorem (the first deep theorem in point-set topology): Every regular space X with a countable basis is metrizable.
2. Ascoli's theorem: Let X be a space a let (Y,d) be a metric space. Give C(X,Y) the topology of compact convergence; let F be a subset of C(X,Y), then
a) If F is equicontinuous under d and the set F_a = {f(a) | f in F} has compact closure for each a in X, the F is contained in a compact subspace of C(X,Y)
b) the converse holds if X is locally compact Hausdorff

I've only done point-set; from what I got out of it was to take concepts from analysis like convergence, open/closed sets, metric spaces, etc & apply them to sets other than the real or complex numbers. Algebraic topology is a kind of extention of geometry where what you want to do is to find out what shapes/figures are the "homeomorphic" (= if there is a continuous bijection from one figure to the other)

From what I keep hearing, the ultimate topology text is the one by Dugundji (Allyn & Bacon, 1960s) but I've never looked at it; it's never in the library. The textbook for the course I did was (of course) the one by Munkres, which was good; the one by Kelley & the one by Willard (<- Canadian content!) are real good too.

 

[matt grime]

Aren't two topolgies the same if they have the same basis? Also, topolgies can be judged to be coarser or finer or noncomparable than each other by looking at their open sets.

Prove or disprove:

all simply connected compact 4 manifolds are homeomorphic to S^4.

when yo'uve got an answer, let me know.

That one is fairly easy as it happens. When you're done try the poincare conjecture, or the geometrization conjecture. One more I'd be interested in is if two quasicoherent sheaves on an algebraic variety have the equivalent derived categories what can you tell me about the underlying variety, particulary if it is BG for some G.

 

[sparkster]

Prove or disprove:

all simply connected compact 4 manifolds are homeomorphic to S^4.

when yo'uve got an answer, let me know.

That one is fairly easy as it happens. When you're done try the poincare conjecture, or the geometrization conjecture. One more I'd be interested in is if two quasicoherent sheaves on an algebraic variety have the equivalent derived categories what can you tell me about the underlying variety, particulary if it is BG for some G.

I had said that I was just getting into topology. All you had to do was say, "No, that actually isn't right."

 

[matt grime]

but what you said is true, it just isn't very useful unless you know a priori that the spaces are obviously in some bijective correspondence, ie have the same (isomorphic as sets) underlying set.

here's an example from Riemann surfaces. Every torus with analytic atlas is realized as a quotient of C by a lattice. These are isomorphic as sets, homeomorphic as topological spaces, even as manifolds, but they aren't isommorphic as analytic manifolds.

Any Lattice is given by two non-colinear complex numbers (ie whose ratio is not real). They are equivalent as Riemann surfaces only if the ratios are the same.

 

[fourier jr]

check this out:
A Hausdorff space X is a topological space in which for each pair x_1, x_2 of distinct points of X there exist neighbourhoods U_1, U_2 of x_1, x_2, respectively, that are disjoint. So in spaces which aren't Hausdorff a sequence can converge to multiple limits, and the limits can be anything!
example for devious:
just take the constant sequence {b_n}. That converges to b, but it also converges to a and also to c, lol. Pretend that x_n is a sequence of points of X that converge to x. If x doesn't equal y let U & V be disjoint neighbourhoods of x & y, respectively. Since U contains x_n for all but finitely many values of n, the set V cannot. So x_n cannot converge to y.

 

[matt grime]

It should be noted more clearly that Fourier_jr is using the topological definition of convergence. A point x is a limit of the sequence x(n) if for any open set, O, containing x, all but finitely many of the elements of x(n) are in O.

 

[sparkster]

but what you said is true, it just isn't very useful unless you know a priori that the spaces are obviously in some bijective correspondence, ie have the same (isomorphic as sets) underlying set.

here's an example from Riemann surfaces. Every torus with analytic atlas is realized as a quotient of C by a lattice. These are isomorphic as sets, homeomorphic as topological spaces, even as manifolds, but they aren't isommorphic as analytic manifolds.

Any Lattice is given by two non-colinear complex numbers (ie whose ratio is not real). They are equivalent as Riemann surfaces only if the ratios are the same.

As I indicated before, I'm taking my first topology class right now. So let me ask question--so far, we've only done point set topology. How do you move from abstract sets with topologies on them to talking about shapes?

That is, how do you go from talking about order topology or k-limit topology on the real line to talking about a taurus?

It should be noted more clearly that Fourier_jr is using the topological definition of convergence. A point x is a limit of the sequence x(n) if for any open set, O, containing x, all but finitely many of the elements of x(n) are in O.

Is every metric space-type topology Hausdorff?

 

[matt grime]

That is, how do you go from talking about order topology or k-limit topology on the real line to talking about a taurus?


Is every metric space-type topology Hausdorff?

You just start talking about them. The torus (taurus is a bull) can come equipped with one of many topologies.

The natural one is the quotient topology from R^2. Then there is the subspace topology inherited from R^3 with the torus considered as embedded in R^3. There is also the realization of the torus as S^1xS^1, the product of two circles, which has the product topology on it. These are all homeomorphic, and here you can use the basis argument to show it. Take one basic open set in one topology, and show it's open in each of the other sets.




Here's another exercise example:

given R, define a set to be closed if it is the set of zeroes of a polynomial. the open sets in the topology are then the complements of closed ones. We won't prove this is a topology, you can take that on trust. It is called the zariski topology.

R^2, also has a topology where you take the closed sets as the sets of zeroes of a polynomial in two variables, again called the zariski topology.

R^2 with this topology is not homeomorphic to the topological space given by the product topology on RxR.

nor is the zariski topology on R equivalent to the metric topology on R. Try thinking how to prove that, let me know how you'd start. it's a useful exercise.
The simplest way to show how this works is to consider R^2 with the metric topology, and with the product topology from RxR, where each R has the metric topology on it, call these topologies T1 and T2.

A basic open set in T1 is simply a little open disc, in T2 it as a little square region (without its boundary). since around every point in an open disc you can draw an open square wholly contained inside the disc, and conversely around every point in an open sqaure you can draw an open disc wholly contained in the open square, the open sets agree and the topologies are the same.

What is it that you particularly want to talk about with respect to topologies on the torus.

Metric topologies are hausdorff (via the triangle in equality)
if x and y are distinct, let d = d(x,y), then th open balls of radius d/3 about each point are open and disjoint.

 

[selfAdjoint]

Is every metric space-type topology Hausdorff?

Sure! And you should be able to prove it yourself. Think of the axioms for a metric. What is a "neighborhood" in a matric space? Now think of the definition of Hausdorff. Now put them together and fiddle around a little bit.