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

Topology appears to be about the study of shapes and whether two objects are the same.

It seems to be about this transformation called a homeomorphism, I've come across explanations that say two things are topologically equivalent (which I suppose means roughly that they "go in" the same place, or class of objects), if there is a homeomorphism or continuous map between them, and you see the 'canonical' example of a coffee cup being deformed into a doughnut or torus. (Hence, they are the same, topologically speaking.)

What about knot topology? That seems to be about deciding whether two knots are the same knot, and possibly mirror images. Knots, as one dimensional loops, and links of knots, are objects that can only exist in three or more dimensions (you can't cross a loop of string over itself in two dimensions). The shadow of a knot is an object which looks like a graph, and so on.

A graph can be just some set of points, all separated, with no edges in it; it can look a lot like a set with no structure imposed on it.

 

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Interesting question! I wonder what the minimal and absolute descriptors for a knot would be. I think it would be in terms of crossovers but I'd like to ponder it more...

 

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In fact, although the term is used in 2 different ways, this is not one of them.

1. Topology is the study of topological spaces, which leads us to....

2. Suppose a point set \(S\) and its powerset which always exists and written, say, \(\mathcal{P}(S)\) one says that the subset \(\tau \subseteq \mathcal{P}(S)\) is a topology on \(S\).

The notation \(S(\tau)\) is then used to denote a topological space. For any given topology thus defined one has that

1. Finite intersection of elements in \(\tau\) (sets, recall) are in \(\tau\)

2. Arbitrary union of elements in \(\tau\) are in \(\tau\)

3. \(S \in \tau\) and \(\emptyset \in \tau\)

The elements in \(\tau\) are accordingly called the open sets in our topological space

The closed sets in our space are simply those sets that are the complement of some open set

It is easy to see that both \(S\) and \(\emptyset\) are both open and closed - there may be others with this property. But if not, one says our space is "simply connected". This is called a "topological property" - there are others we can talk bout if you want.

For any given set, there may be many different topologies available, but usually (not always) one doesn't care which one it is and simple writes "\(X\) is a topological space".

So, finally, given topological spaces \(X,\,\,Y\) and a mapping \(f:X \to Y\), one says that this is continuous if, for every open set \(V \in Y\) its pre-image set \(f^{-1}(V)\) is open in \(X\). And if this mapping has an inverse, similarly defined, one calls this a homeomorphism.

Homeomorphisms preserve topological properties, in which case one says that \(X\) and \(Y\) are "topologically equivalent"

I suggest you understand all this (and more) before proceeding to knot theory

 

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Or...you know...we can skip all the bloviating and discuss knot theory here.

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That's a misuse of the term "bloviating."

Foundations, such as concrete definitions and logical development or even a capsule synopsis of the history of logical development, are essential to mathematical discussion.

 

Why, sure, go ahead if you are truly competent to do so

 

The word was intentional. QH's technical definition does little to foster discussion IMO. My interest is piqued though so I'll do some thinking and a bit of reading on the subject. At first pass it looks like knot theory only deals with connected ends (basically continuous loops)...I'm not sure this is equivalent to the subject as I understood it but if I get time I'll do some research

 

Foundational Lecture Notes on Point-Set Topology
Cribbed from metamath.org 's introductory theorems and definitions.

A “topology”, τ, is a set of mutually-related “open sets.” These open sets have the properties with respect to their topology.
1) The union of ANY subset of τ is itself an open set of τ.
2) The intersection of any two “open sets” of t is itself an open set of τ.

IsTopology(τ) ↔ ( ( ∀ x ∈ powerset(τ) [ union(x) ∈ τ ] ) AND ( ∀ x, y ∈ τ [ (x ∩ y) ∈ τ ] ) )

Thus the empty set is an open set of τ because the empty set is the union of 0 subsets of τ.
Likewise there is a smallest set of which every open set of τ is necessarily a subset, which is the union of all of τ. Since all of τ is the largest subset of τ, the union of all of τ is necessarily an open set of τ.

3) ∅ ∈ τ
4) union(τ) ∈ τ

Since the union of all open sets is itself an set, then we may introduce the “base set” of a the topology. Talking about a specific pair of base set and topology we are taking about a “topological space” where the elements of the base set are generally called “points”.

GenerateBaseSet(τ) := union(τ)

IsTopologicalSpace(<T, τ>) ↔ ( IsTopology(τ) AND T = GenerateBaseSet(τ) )

And the intersection of any non-empty and finite subset τ is also of an open set of τ. ( This follows from finite induction on the size of the subset, and does not extent to infinite subsets of τ. )

A “topological basis” is a set, b, with the property that for any two elements of b, x and y, their intersection z is a subset the union of all subsets of z which also are elements of b.

Given a topological basis, b, we may form the set of all sets, o, that are subsets of the union of all subsets of o which are also elements of b. This set of all such open sets generated by the basis b, is a topology.

Expand(o, b) = union( b ∩ powerset(o) )

IsTopologicalBasis(b) ↔ ∀ x, y ∈ b [ (x ∩ y) ⊆ Expand((x ∩ y), b) ]

GenerateTopology(b) := { x | x ⊆ Expand(x, b) }

//Edit -- meant to add this at end:
... belatedly I now believe that QuarkHead was challenging RJBeery to discuss knot theory, but I maintain that Topology should be the primary topic of this thread.

 

Crossed posts with rpenner. I confess I had to look up "bolviating", a term unknown in the UK.

I thank him for his (apparent0 support.

Readers, let me get this off my chest..... yes rpenner and I disagreed about something relatively minor in diff.geom. So what? Everything is up for interpretation, even in pure mathematics.

This is no way way diminishes the fact that I hugely respect his mathematical knowledge and abilities. I would hope it is mutual, at least to some degree

 

no.

 

Yes
krash, I thank you for this informed and stimulating discussion

 

Ok, well I'm happy to discuss what Topology is, but I'm also wanting to discuss particular examples like knot and graph topologies. What kind of open sets are there in either case, or what is a good way to describe the topological basis of knots?
From http://en.wikipedia.org/wiki/Topology:

.
Are there issues with this description, given what's been posted so far?

So a graph with edges has this connectedness property, whereas the length of any edge isn't necessarily a property--all the edges are deformable; in knot theory you can deform a knot but it's the same knot (for instance you can stretch part of a trefoil knot and this doesn't change it).

 

Ahh, check out the HOMFLY polynomial. It looks like it breaks down knots into their mathematical roots based on crossovers. Cool stuff

Also here's the best summary I've found:

http://www.math.buffalo.edu/~menasco/Knottheory.html

 

Once again, I urge you to walk before you try to run. BTW, the Wiki article you cite is misleading in that it fails to distinguish between point-set topology and algebraic topology

OK, "Barkis is willing", so....

A topological space \(X\): is said to be connected iff it cannot be written as the union of two non-empty disjoint sets. The intuitive content here should be clear: if \(X = A \cup B\), and \(A \cap B = \emptyset\), then I cannot "move" from a point in A to a point in B without falling into a chasm.

This is, in fact, a rather antiquated (but perfectly serviceable) definition. The better definition is as follows: a topological space \(X\) is said to be connected iff the only subsets of \(:X\) that are both open and closed are \(X\) and \(\emptyset\). This is the definition I gave earlier

These two definitions are easily brought into register: Let \(A \subsetneq X, \, B \subsetneq X\) be open. Let \(X = A \cup B\), and let \(A \cap B = \emptyset\) (recall this is the the definition of disjointness). Then of necessity, \(B = A^c \in X,\, A = B^c \in X\), (where \(A^c\) is the complement of the set \(A\), as I defined it earlier) therefore A and B are both open and closed, and \(X\) is not connected by either definition.

The other topological property I want to mention is compactness. Again, there are two parallel definitions, and I offer the oldest, and most intuitive, first.

A set \(A\) is said to be compact iff, for every sequence in \(A\) that has a limit, that limit is found in \(A\).

Again, the content should be clear: a compact set is "self-contained" with respect to limits.

The grown-up version of the same property will require a bit of a detour, which I am going to leave for now. Likewise the so-called "separation axioms", another topological property

Let me know if the above is less than clear

 

For the more visually minded, I have uploaded the trivial topology for a set of three items and the discrete topology for the same three items. Here the points are solid discs and the open sets are circles. The central tiny circle is the empty set, depicted here as a subset of all other open sets.

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There are 64 possibilities which are intermediate between these two extremes of which only 27 qualify as topologies.

Because it is the nesting properties of the circles that make it a topology, any number of additional points could be added between the black circles (but not on the circles themselves) and the resulting relationship would still be a valid topology over the larger base set.

QuarkHead's much more likely to talk about topologies where the number of open sets is larger than the set of counting numbers. I won't feel like diagramming too many of those.

 

A couple of observations about knots:

The shadow of every knot crossing is a 4-valent vertex in a graph, so knots represent a subset of all graphs.
Knot diagrams are graphs with extra structure, which is the breaks in some part of a loop where these undercross at a vertex (normal graphs don't have these diagrammatic structures). This means the crossings and hence the knot, can be oriented, unlike a graph.

 

Not I!! It can be done, as follows....

Consider the natural (counting) \(\mathbb{N}\) as a set. Then there is a lovely proof of Cantor (the so-called "diagonal argument") that shows that the powerset \(\mathcal{P}(\mathbb{N})\) is uncountable. And since \(\tau = \mathcal{P}(\mathbb{N})\) is a valid topology on the set of counting numbers, then as you say, the number of open sets is uncountable.

In fact for any set \(S\) one has that \(\tau= \mathcal{P}(S)\) - this is called the "discrete topology". But it is somewhat pathological, and never (as far as I know) used

But let me continue with our topological properties. Recall I said a set is compact if it contains all its limits. This in fact is known as "sequentially compact". Here's a more general notion of compactness.

Let \(X\)be a topological space. I suppose a class (it's probably too big to be a set) of sets in \(X\) whose union is all of \(X\). Then I call this a covering (or cover) for \(X\) - this cover always exists. And guess what - if all these sets are open, I call this an open cover.

And if there is a finite sub-set of this open cover that is still a cover, then by definition \( X\) is compact.

And if each set in this finite subset has a subset which is still aan open cover, I will say that \(X\) is paracompact.

These are obviously quite stringent requirements, but, together with the other topological properties I introduced, serve the useful purpose of weeding out some of the more beastly top. spaces that may arise merely from the bald definitions.

One more property - the separation axioms. But later for that

 

arfa: topology and knots are a big deal because particles are essentially knots. See what rpenner said here:

"So in summary, a quantum bound state is a little like a high-dimensional knot of quantum field configurations that happens to be an energy eigenstate which says the state doesn't change over time in any fundamental way."

Take a look at the Topological Quantum Field Theory Club webpage. See those blue trefoil knots at the top? Pick one, start at the bottom left, and trace around it anticlockwise calling out the crossing-over directions: up down up.

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Good God, is there no getting rid of this pest? Farsight, please demonstrate at least SOME understanding of the subject at hand.

Remember you are the guy trained in "analysis and logic". Use it, or shut the feck up

 

I understand the subject. You've said repeatedly that you're not a physicist, and we have free speech in science, so back off. Particularly since the last time you tried to give a lesson it was useless, and rpenner shot you down:

http://www.sciforums.com/threads/manifolds-tensors-and-fields.143062/