What is Topology, and what's the big deal?

Incredible. You think that rpenner "shot down" Quarkhead?

I can't help but think you're deliberately trolling, because no one can be that stupid.

 

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Oh really?? I think may have made a serious mistake in my last post. Please correct it for me.

So what? I am a mathematician (of sorts) and my "started threads" are on this subject alone

Well rpenner can speak for himself, but I do not feel "shot down" - he and I merely disagreed on a small matter of interpretation. Maybe you would like to explain to us how YOU would describe, mathematically, a coordinate transformation on the manifold as I defined it

 

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Here's one of my favorite somewhat obscure beastly examples.

It's well-known that if a space is compact, every continuous function from that space to the real numbers is bounded.

What about the converse? If every continuous function from a space X to the reals is bounded, must X be compact? The answer is no. Such a space is called pseudocompact.

An example of a space that's pseudocompact but not compact is any infinite space with the particular point topology. That's the topology in which you pick some arbitrary point, call it \(x\), and then say that the open sets are any set containing \(x\), along with the empty set. You can see that such a collection of sets is closed under arbitrary unions and finite intersections (arbitrary intersections in fact), and that the empty set and the entire space are in the collection. So this is a topology.

Consider a space \(X\) whose points are the natural numbers 1, 2, 3, 4, ... with the particular point topology on the number 1. This space is not compact, because if I take the collection {1,2}, {1,3}, {1,4}, {1,5}, ... this is an open cover of \(X\); but any finite subcollection does not cover \(X\). So \(X\) is not a compact space.

On the other hand, any continuous function from \(X\) to \(\mathbb{R}\) is bounded. To see this, suppose that \(f : X \rightarrow \mathbb{R}\) is continuous. If \(f\) is a constant function it's bounded, so suppose \(f\) is non-constant. Then its image contains two distinct reals \(a\) and \(b\), with two disjoint open neighborhoods \(A\) and \(B\) containing \(a\) and \(b\), respectively. That's a consequence of the usual topology on the reals, in which any two distinct points can be separated by epsilon-neighborhoods. Moreover, \(A\) and \(B\) are each nonempty.

A function on a topological space is continuous if and only if the inverse image of any open set is open. (It's a nice exercise to show that this definition is equivalent to the usual epsilon-delta definition from calculus).

Therefore the inverse images \(f^{-1}(A)\) and \(f^{-1}(B)\) are nonempty disjoint open sets in \(X\). But this is impossible, because there are no two nonempty open sets in \(X\) that are disjoint; their intersection always contains at least the number 1.

Therefore \(f\) is constant after all. In other words, every continuous function from \(X\) to \(\mathbb{R}\) is constant, hence bounded. But \(X\) is not compact.

http://en.wikipedia.org/wiki/Pseudocompact_space

http://en.wikipedia.org/wiki/Particular_point_topology

 

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My understanding of the subject, poor as it may be, is that it started with Euler and the Bridges of Konigsburg problem; this is a problem which is firmly set in graph theory. It wasn't until the 20th century that topological spaces came along, which I understand as the most general, or foundational, formalism.

It should be obvious that a graph's topology is quite different to that of a fixed kind of geometrical object like a sphere or torus. It's also a little difficult for me to see where the notion of a continuous function between graphs comes into the frame.

 

Me too, but we keep on learning. Some damned good posts in this thread, and thanks.

 

Then take a look at Euler's own generalization of the Konigsburg problem.

He states that for any graph, that (as numbers of) Vertices + Exterior faces - Edges = 0 or 2 (these are known as the "Euler characteristic". Often written as \(\chi\)) No other possibilities exist.

To make contact with what you call "geometric objects" - consider all possible polygons inscribed exactly in the circle - a 1-dimensional manifold. Since there are no exterior faces one has that \(n+0-n = \chi=0\) Try as an experiment the inscribed triangle. This generalizes

Likewise try as a simple example the 4-cube inscribed exactly inside the 2-sphere. There are 8 vertices + 6 exterior faces - 12 edges = 2. This generalizes.

So all geometric objects (so-called by you) have Euler characteristic zero or two, which you may care to try and prove (or disprove.)

But this is a far cry from the sort of introductory topology we are talking here so far. Sorry to have brought your speculations to an earthly "bump"

 

and you have said the same thing many of times.
seriously ? you had to contaminate this topic also ?

 

Can you channel Martin Gardner?

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I like this, thank you very much.

 

It's all my fault, Dr. Toad.

I had been searching for this document for decades (literally) and finally found it here:

http://projecteuclid.org/euclid.cmp/1104162092

For all to see. Notice that its author, Dr. Edward Witten, noted high energy physicist from Princeton who pieced together M theory (now Brane theory) proposed this as a topological description of interaction with the sigma field.

Notice how it has a very prominent section on gravity. You already know all about how it bothers me when someone explains spooky action at a distance by a description of curvature of space as though space were a simple rubber sheet or a child's sliding board. That model will work fine up to a point, but we are long past the time when that is an adequate explanation to advance physics.

If you complain to such folks about how did it go from their math notebooks to God's master plan, they will explain that to you. But they will never explain why it is they believe that an interaction that happens between solid matter and a bosonic superfluid is going to be explained by means of the same math that explains doughnuts, Klein bottles and Mobius bands, all of which are solid. Gravity is evidently not about interaction between solids alone. That idea simply does not work. If the mathematical model does not explain how inertia is imparted to matter from the vacuum and how this occurs continuously, more than a small part is missing.

I know a poor application of math when I see it. If this doesn't jump out at you as something that is modeled using the wrong math, I don't know what does.

And they've been building on this flawed model for the last 30 years. Lots of purely mathematical inertia there, apparently.

 

I'm not sure I understand "Exterior faces" in the context of graphs.

A graph has faces if it's planar, which means no pair of its edges intersect; if it's planar it can be drawn on a sphere. And so if a graph isn't planar, "faces" I guess is undefined; (or maybe the graph fits on a torus or some more exotic surface).

And I've been taught that a planar graph has one more "face" than those areas inside the diagram, this extra face is the plane (resp. the rest of the sphere).

Further, for convex polyhedra Euler's formula gives char 2, but there are examples of non-convex polyhedra that have char 0, 1, -2 (on this page).

 

arfa: re what I was saying, note how rpenner's depiction in post #15 above somewhat resembled the proton on Wikipedia:

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And of course, when it comes to TQFT, one notes that the trefoil is tricolourable:

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Again the crossing-over points can be read out as up down up. IMHO it's important to bear this in mind when thinking about why we have never seen a free quark. Topology isn't just some arcane branch of mathematics, IMHO it is a big deal. I'd go so far as to say it's the difference between light and matter.

Dan: gravity is easy, but that's a different subject.

 

Let me just finish my list of the most important properties, that of separation. Basically this has to do with what criteria we use to decide whether 2 points are the same of different. This may sound like a silly thing to worry about, but we may not have a metric atour disposal, so using the obvious \(|x-y|=0 \Rightarrow x=y\) won't work.

These criteria are given by the so-called "separation axioms" and go by the loveable names \(T_0,\,T_1,......,T_5\).
The only one of these that is not in some sense bizarre or even nuts is \(T_2\), which goes as follows.

Suppose \(X\) a topological space with \(x,\,\,y \in X\). Then if I can never find open sets in \(X\) such that \(U \ni x \cap V \ni y= \emptyset\) then I am obliged to say \(x=y\)

Conversely, if I can always find such disjoint open sets then \(x \ne y\).

Note that the qualification "open" is vital - the singleton sets \(\{x\}\) are closed in all topologies known to man.

A topological space satisfying this \(T_2\) separation axiom is called a "Hausdorff space".

So the homeomorphic image of a connected space is connected, the homeomorphic image of a compact space is compact, the homeomorphic image of a Hausdorff space is Hausdorff and gives rise to the notion of 2 (or more) spaces being topologically equivalent.

 

Part 3.

Symbols previously used:
τ — A “topology” (technically, this is a point-set topology, as other definitions exist), where both the union of any subset of τ is an element of τ and the intersection of any two elements of τ is an element of τ.
o — An “open set”, an element of a particular topology
T — The “base set” of a “topological space”, the union of all open sets. Elements of T are called “points”
<T, τ> — A topological space is a pair of base set and a certain topology.
b — A “topological basis” where the intersection of any two elements of b is a subset (or equal to) the union of all subsets of that intersection which are also elements of b.

I also gave an examples of the “trivial topology” where τ contained only the empty set and the whole of T and of the “discrete topology” where every subset of T was an open set.

Why did we introduce the term topological space if the topology has the same information? IsTopologicalSpace(<union(τ), τ>) ↔︎ IsTopology(τ), so why the extra notation? My sources don't speak to that but I think it's partly because the theme of a base set + structure + properties is a paradigm in how mathematicians think about things. It's less clumsy to think of points p, q as elements of T rather than union(τ) and it's easier to talk about base sets with more than just one type of structure.

Given a topological space we may also construct the set of “closed sets”, τ̅, which is defined as τ̅ = {c | o ∈ τ AND c = T \ o }. Thus every closed set is a complement of an open set. However, in point set topology, unlike English, the same set may be both an open set and a closed set. τ ∩ τ̅ ≠ ∅ Depending of the topology, some subsets of the base set may be neither open or closed. Because the set of all closed sets has the same information content as the set of all open sets, it is not surprising that De Morgan's laws can be used to define topology entirely in terms of closed sets.

We can define the closure of a subset of the baseset in a particular topological space to be the smallest closed set which holds the open set as a subset. We can make this slightly vague concept concrete as the complement of the union of all open sets which are disjoint from our chosen subset. By De Morgan's law, the closure of a subset, S, is equal to the intersection of the set of all closed sets which contain it.

Closure(S, <T, τ>) := T \ union({ x | x ∈ τ AND S ∩ x = ∅ }) = intersection({x | ( T \ x ) ∈ τ AND S ⊆ x ⊆ T })

Naturally, the closure of a particular closed set is that closed set, Closure(S, <T, τ>) = S if and only if ( T \ S ) ∈ τ.

For both the trivial topology and the discrete topology, τ = τ̅, which makes the closure of an open set trivial, o = closure(o, <T, τ>).

I hope to return to closed sets in the future.

For any subset of the base set, S ⊆ T, we can define the interior of that set as the union of subsets of the subset that also happen to be open sets.

Interior(S, <T, τ>) := Expand(S, τ) = union( τ ∩ powerset(S) )

Much like open and closed are related by set complements, so are Interior() and Closure(). The interior of a particular open set is that open set, Interior(S, <T, τ>) = S if and only if S ∈ τ. T \ Closure(S, <T, τ>) = Interior(T \ S, <T, τ>).

∅ ⊆ Interior(X ∩ T, <T, τ>) ⊆ (X ∩ T) ⊆ Closure(X ∩ T, <T, τ>) ⊆ T.
Closure(S, <T, τ>) = Closure(Closure(S, <T, τ>), <T, τ>).
Interior(S, <T, τ>) = Interior(Closure(S, <T, τ>), <T, τ>).

Closure and Interior capture and generalize a property of the border of geometric figures, if a point is on the border of one figure and the interior of another, then the two figures have non-disjoint interiors. S₁, S₂ ⊆ T AND Closure(S₁, <T, τ>) ∩ Interior(S₂, <T, τ>) ≠ ∅ → Interior(S₁, <T, τ>) ∩ Interior(S₂, <T, τ>) ≠ ∅.


A subset of a topology is a basis for the topology if and only if every member of the topology is a union of members of the basis. If b is a topological basis and τ is the topology generated from that basis, then they have the same union or base set. union(b) = union(τ) = T.

If τ₁ and τ₂ are topologies on the same base set, the if τ₁ ⊆ τ₂, then we say τ₁ is a “coarser” topology than τ₂ or that τ₂ is a “finer” topology than τ₁. If the relationship is τ₁ ⊂ τ₂ then we say “strictly coarser” or “strictly finer”, respectively.

IsCoarserTopology(τ₁, τ₂) := union(τ₁) = union(τ₂) AND τ₁ ⊆ τ₂

If b₁ and b₂ are topological bases and τ₁ and τ₂ are their respective topologies then some relations between b₁ and b₂ describe relations between τ₁ and τ₂.
For example, b₁ ⊆ b₂ implies τ₁ ⊆ τ₂. (But note that b₁ ⊂ b₂ does not imply τ₁ ⊂ τ₂. ) And if b₁ and b₂ are topological bases on the same base set then saying "for all x in in b₁ the intersection of x with the base set of b₁ is a subset of (or equal to) the union of intersection of b₂ and the powerset of x" is logically equivalent to τ₁ ⊆ τ₂.

IsCoarserBasis(b₁, b₂) := union(b₁) = union(b₂) AND ∀o ∈ b₁ [ ( union(b₁) ∩ o ) ⊆ Expand(o, b₂) ) ]

IsCoarserBasis(b₁, b₂) ↔ IsCoarserTopology(τ₁, τ₂)

Obviously the trivial topology is the coarsest possible topology on any base set, while the discrete topology is the finest.

Early Examples of point-set topologies:
• The Trivial topology is sometimes called the indiscrete topology, because if p and q and points of the topological space, there is no open set that have p but not q.
• The Discrete topology means the closure of any subset is itself. That's almost no topological concept at all, like a bag of unrelated things.
• A topology can induce a topology on any subset of its base set. This is called the subset topology and its open sets are just the intersections of the open sets of the parent with the base set of the child.
• For any base set, there is the finite complement topology where the non-empty open sets consist of all complements of finite sets.
• Likewise, the countable complement topology where the non-empty open sets consist of all complements of countable sets.

 

I confess I don't quite understand why rpenner repeats all I have said in this thread, using "programmer's language" and a different notation. I dare say he has good reason

Anyway, I see now I missed out something important, but since I started wittering about Hausdorff spaces, I had better finish. Let me write \(x \in U \equiv U_x\) for a neighbourhood of this point

If \(U_x \cap U_y = \emptyset\) and \( U_y \cap U_z = \emptyset\) this does not imply that \(U_x \cap U_z = \emptyset\), i.e this property is not transitive.

However, Hausdorff asserts that, nonetheless, there will be neighbourhoods \(V_x \,V_z \) such that \(V_x \cap V_z = \emptyset\).

Let me go back to the bit that should have followed the introduction of closed vs, open sets. Here, our intuition will be a reliable guide, provided we keeps our heads.

Again let \(X\) be a topological space with \( A \subseteq X\) (notice I haven't specified whether it is open or closed)

Then the interior of \(A\) is the largest open set contained in \(A\), and is written \( A^o\). Equivalently, we may say that \( A^o\) is the union of all open sets in \(A\). Evidently \(A^o\) is open in \(X\), and if \(A\) is open, \(A = A^o\), not otherwise..

Conversely (in a manner of speaking), the closure of \(A\), written \(A^-\), is the smallest closed set containing \(A\). Equivalently, \(A^-\) is the intersection of all closed sets of which \(A\) is a subset. Equally evidently, \(A^-\) is a closed set, and if \( A\) is closed, \(A = A^-\), not otherwise..

I now define the boundary \(\partial A\) of \(A\) as \( \partial A = A^- \setminus A^o\). Note the notation "\(\setminus\)" is the set-theoretic equivalent of the arithmetic "minus" and the \(\partial\) in the definition of the boundary has nothing whatever to do with derivatives

Note also that the boundary of a set thus defined is not the same as its bounds (though it is related)

This is merely a fancy way of saying that the boundary of a closed set is included in the set, the boundary of an open set is not.

The demonstration that this must be true is easy.

 

I thought that topology is the study of geometric properties which are independent of making measurements.

For example: Topology notes that a sphere & a torus are fundamentally different, while a cube & a sphere are equivalent because either can be transformed into the other by continuous deformations which do not require cutting either object.

 

Well, I guess this has gone about how you'd expect.

QuarkHead and rpenner have presented some formal definitions, these seem to be about open sets and how this "openness" is defined.

What if, I was thinking, the set is a permutation group, or the set is a set of group elements? In that case some subsets will be closed if they form subgroups under the group operation (which is composition of permutations). Does this give some group, say the symmetric group on three elements, as in rpenner's example, a topological structure?

 

I don't think a group gives a topology. I think a base set can be given simultaneously the structure of a group and an topology. This suggests to me that the group action would always map open sets to open sets and that the open sets would each have subgroups for which they were invariant. So not any group action would work with any topology.

 

If a group happens to have a topology *and* the group operation and inverse operations are continuous under that topology, then you have a topological group, which is an object of interest in math.

http://en.wikipedia.org/wiki/Topological_group

 

I'm still trying to visualise open sets and set relations in respect of knots or graphs. Is a graph a topological space, and where do De Morgan's Laws come into it?

 

A graph can be used to derive a topological space. For example, if each edge is a copy of the open real number interval (0,1) and it's two endpoints are in the closure of that open set, you have a basis for a natural "graph topology" .

But the topology of graphs is quite different from the topology of knots and links because what makes knots knotty is that they are embedded in a manifold (typically Euclidean space) which is why knots differ from the unknot even though knots and the unknot would just be topologically equivalent to circles if it weren't for the embedding. Since topological knot theory winds up being the topology of a manifold with a peculiar winding hole cut from it a good basis in foundations of topology would seem to be necessary to make headway in or even to make sense of knot theory.

I think it is likely that visualization is something that doesn't play a big role in this area as knots scale up larger than human spatial modeling system can easily handle. (c.f. Gordian Knot)